The formal definition of limits Calculus without Limits C. K. Raju Introduction d ◮ On present-day mathematics, the symbol dx is defined The difficulty of for a function f : R → R , using another symbol lim h → 0 . limits The difficulty of ◮ defining R f ( x + h ) − f ( x ) df dx = lim . The difficulty of h set theory h → 0 The integral ◮ lim h → 0 is formally defined as follows. The difficulty of defining functions x → a g ( x ) = l lim What the student takes away if and only if ∀ ǫ > 0 , ∃ δ > 0 such that Calculus with limits: why teach 0 < | x − a | < δ : ⇒ | g ( x ) − l | < ǫ ∀ x ∈ R . it? Conclusions Appendix
The missing element Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a section The difficulty of set theory called “precise definition of limits”. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The missing element Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a section The difficulty of set theory called “precise definition of limits”. The integral ◮ But the definitions given are not precise. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The missing element Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a section The difficulty of set theory called “precise definition of limits”. The integral ◮ But the definitions given are not precise. The difficulty of defining ◮ They have the ǫ ’s and δ ’s. functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The missing element Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a section The difficulty of set theory called “precise definition of limits”. The integral ◮ But the definitions given are not precise. The difficulty of defining ◮ They have the ǫ ’s and δ ’s. functions ◮ But are missing one key element: R . What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory The difficulty of defining R The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The formal reals Calculus without Limits Dedekind cuts C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory ◮ Formal reals R often built using Dedekind cuts. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The formal reals Calculus without Limits Dedekind cuts C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory ◮ Formal reals R often built using Dedekind cuts. The integral The difficulty of ◮ or via equivalence classes of Cauchy sequences in Q . defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Imitating the European experience Calculus without Limits C. K. Raju Introduction ◮ Teaching R is regarded as too complicated and is The difficulty of limits postponed to texts on advanced calculus 4 or The difficulty of mathematical analysis. 5 defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Appendix Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Imitating the European experience Calculus without Limits C. K. Raju Introduction ◮ Teaching R is regarded as too complicated and is The difficulty of limits postponed to texts on advanced calculus 4 or The difficulty of mathematical analysis. 5 defining R The difficulty of ◮ Notice that this repeats the European historical set theory experience. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Appendix Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Imitating the European experience Calculus without Limits C. K. Raju Introduction ◮ Teaching R is regarded as too complicated and is The difficulty of limits postponed to texts on advanced calculus 4 or The difficulty of mathematical analysis. 5 defining R The difficulty of ◮ Notice that this repeats the European historical set theory experience. The integral The difficulty of ◮ Calculus came first, the ǫ – δ definition of limits followed, defining functions and then R was constructed. What the student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Appendix Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Imitating the European experience Calculus without Limits C. K. Raju Introduction ◮ Teaching R is regarded as too complicated and is The difficulty of limits postponed to texts on advanced calculus 4 or The difficulty of mathematical analysis. 5 defining R The difficulty of ◮ Notice that this repeats the European historical set theory experience. The integral The difficulty of ◮ Calculus came first, the ǫ – δ definition of limits followed, defining functions and then R was constructed. What the ◮ (Cauchy 1789-1857, Dedekind 1831-1916) student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Appendix Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The problem of set theory Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ The construction of R requires set theory. The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The problem of set theory Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ The construction of R requires set theory. The difficulty of defining R ◮ Students are not taught the definition of a set. The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The problem of set theory Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ The construction of R requires set theory. The difficulty of defining R ◮ Students are not taught the definition of a set. The difficulty of set theory ◮ What the student typically learns is something as The integral follows. The difficulty of defining “A set is a collection of objects” functions What the or student takes away “A set is a well-defined collection of objects” Calculus with limits: why teach it? Conclusions Appendix
What set theory the student learns Calculus without Limits C. K. Raju ◮ With such a loose definition it is not possible to escape Introduction things like Russell’s paradox. The difficulty of limits The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix 6 e.g. P. R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
What set theory the student learns Calculus without Limits C. K. Raju ◮ With such a loose definition it is not possible to escape Introduction things like Russell’s paradox. The difficulty of limits ◮ Let The difficulty of R = { x | x / ∈ x } . defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix 6 e.g. P. R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
What set theory the student learns Calculus without Limits C. K. Raju ◮ With such a loose definition it is not possible to escape Introduction things like Russell’s paradox. The difficulty of limits ◮ Let The difficulty of R = { x | x / ∈ x } . defining R The difficulty of ◮ If R ∈ R then, by definition, R / ∈ R so we have a set theory contradiction. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix 6 e.g. P. R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
What set theory the student learns Calculus without Limits C. K. Raju ◮ With such a loose definition it is not possible to escape Introduction things like Russell’s paradox. The difficulty of limits ◮ Let The difficulty of R = { x | x / ∈ x } . defining R The difficulty of ◮ If R ∈ R then, by definition, R / ∈ R so we have a set theory contradiction. The integral The difficulty of ◮ On the other hand if R / ∈ R then, again by the definition defining functions of R , we must have R ∈ R , which is again a What the contradiction. So either way we have a contradiction. student takes away Calculus with limits: why teach it? Conclusions Appendix 6 e.g. P. R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
What set theory the student learns Calculus without Limits C. K. Raju ◮ With such a loose definition it is not possible to escape Introduction things like Russell’s paradox. The difficulty of limits ◮ Let The difficulty of R = { x | x / ∈ x } . defining R The difficulty of ◮ If R ∈ R then, by definition, R / ∈ R so we have a set theory contradiction. The integral The difficulty of ◮ On the other hand if R / ∈ R then, again by the definition defining functions of R , we must have R ∈ R , which is again a What the contradiction. So either way we have a contradiction. student takes away ◮ Paradox is supposedly resolved by axiomatic set theory, Calculus with but even among professional mathematicians, few learn limits: why teach it? axiomatic set theory. Conclusions Appendix 6 e.g. P. R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
What set theory the student learns Calculus without Limits C. K. Raju ◮ With such a loose definition it is not possible to escape Introduction things like Russell’s paradox. The difficulty of limits ◮ Let The difficulty of R = { x | x / ∈ x } . defining R The difficulty of ◮ If R ∈ R then, by definition, R / ∈ R so we have a set theory contradiction. The integral The difficulty of ◮ On the other hand if R / ∈ R then, again by the definition defining functions of R , we must have R ∈ R , which is again a What the contradiction. So either way we have a contradiction. student takes away ◮ Paradox is supposedly resolved by axiomatic set theory, Calculus with but even among professional mathematicians, few learn limits: why teach it? axiomatic set theory. Conclusions ◮ Most make do with naive set theory. 6 Appendix 6 e.g. P. R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory The integral The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction ◮ Most calculus courses define the integral as the The difficulty of limits anti-derivative, with an unsatisfying constant of The difficulty of integration. defining R The difficulty of set theory � if d dx f ( x ) = g ( x ) then g ( x ) dx = f ( x ) + c The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction ◮ Most calculus courses define the integral as the The difficulty of limits anti-derivative, with an unsatisfying constant of The difficulty of integration. defining R The difficulty of set theory � if d dx f ( x ) = g ( x ) then g ( x ) dx = f ( x ) + c The integral The difficulty of defining ◮ It is believed that some clarity can be brought about by functions teaching the Riemann integral obtained as a limit of What the student takes sums. away � b n Calculus with � f ( x ) dx = lim f ( t i )∆ x i limits: why teach it? µ ( P ) → 0 a i = i Conclusions Appendix
The integral Calculus without Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction ◮ Most calculus courses define the integral as the The difficulty of limits anti-derivative, with an unsatisfying constant of The difficulty of integration. defining R The difficulty of set theory � if d dx f ( x ) = g ( x ) then g ( x ) dx = f ( x ) + c The integral The difficulty of defining ◮ It is believed that some clarity can be brought about by functions teaching the Riemann integral obtained as a limit of What the student takes sums. away � b n Calculus with � f ( x ) dx = lim f ( t i )∆ x i limits: why teach it? µ ( P ) → 0 a i = i Conclusions ◮ Here the set P = { x 0 , x 1 , x 2 , . . . x n } is a partition of the Appendix interval [ a , b ] , and t i ∈ [ x i , x i − 1 ] .
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of ◮ and a proof of the fundamental theorem of calculus. set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of ◮ and a proof of the fundamental theorem of calculus. set theory ◮ This is not done. Instead, the focus is on mastering The integral The difficulty of techniques. defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of ◮ and a proof of the fundamental theorem of calculus. set theory ◮ This is not done. Instead, the focus is on mastering The integral The difficulty of techniques. defining functions ◮ the two key techniques of (symbolic) integration are What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of ◮ and a proof of the fundamental theorem of calculus. set theory ◮ This is not done. Instead, the focus is on mastering The integral The difficulty of techniques. defining functions ◮ the two key techniques of (symbolic) integration are What the ◮ integration by parts (inverse of Lebiniz rule), and student takes away Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of ◮ and a proof of the fundamental theorem of calculus. set theory ◮ This is not done. Instead, the focus is on mastering The integral The difficulty of techniques. defining functions ◮ the two key techniques of (symbolic) integration are What the ◮ integration by parts (inverse of Lebiniz rule), and student takes away ◮ integration by substitution (inverse of chain rule) Calculus with limits: why teach it? Conclusions Appendix
The integral Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R The difficulty of ◮ and a proof of the fundamental theorem of calculus. set theory ◮ This is not done. Instead, the focus is on mastering The integral The difficulty of techniques. defining functions ◮ the two key techniques of (symbolic) integration are What the ◮ integration by parts (inverse of Lebiniz rule), and student takes away ◮ integration by substitution (inverse of chain rule) Calculus with ◮ since integration techniques are more difficult to learn limits: why teach it? than differentiation techniques. Conclusions Appendix
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory The difficulty of defining functions The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The difficulty in defining functions Calculus without Limits C. K. Raju ◮ Thus the student learns differentiation and integration Introduction as a bunch of rules. The difficulty of limits The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The difficulty in defining functions Calculus without Limits C. K. Raju ◮ Thus the student learns differentiation and integration Introduction as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary to The difficulty of defining R define functions, such as sin( x ) The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The difficulty in defining functions Calculus without Limits C. K. Raju ◮ Thus the student learns differentiation and integration Introduction as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary to The difficulty of defining R define functions, such as sin( x ) The difficulty of ◮ However, the student does not learn the definitions of set theory sin( x ) etc. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
The difficulty in defining functions Calculus without Limits C. K. Raju ◮ Thus the student learns differentiation and integration Introduction as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary to The difficulty of defining R define functions, such as sin( x ) The difficulty of ◮ However, the student does not learn the definitions of set theory sin( x ) etc. The integral The difficulty of ◮ since the definition of transcendental functions involve defining functions infinite series and notions of uniform convergence. What the student takes ∞ away x n e x = � n ! . Calculus with limits: why teach n = 0 it? Conclusions Appendix
The difficulty in defining functions Calculus without Limits C. K. Raju ◮ Thus the student learns differentiation and integration Introduction as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary to The difficulty of defining R define functions, such as sin( x ) The difficulty of ◮ However, the student does not learn the definitions of set theory sin( x ) etc. The integral The difficulty of ◮ since the definition of transcendental functions involve defining functions infinite series and notions of uniform convergence. What the student takes ∞ away x n e x = � n ! . Calculus with limits: why teach n = 0 it? Conclusions ◮ The student hence cannot define e x , and thinks sin( x ) Appendix relates to triangles.
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory What the student takes away The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
What the student takes away Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, the best that a good calculus text can do is to defining R trick the student into a state of psychological The difficulty of set theory satisfaction of having “understood” matters. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
What the student takes away Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, the best that a good calculus text can do is to defining R trick the student into a state of psychological The difficulty of set theory satisfaction of having “understood” matters. The integral ◮ The trick is to make the concepts and rules seem The difficulty of defining intuitively plausible functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
What the student takes away Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, the best that a good calculus text can do is to defining R trick the student into a state of psychological The difficulty of set theory satisfaction of having “understood” matters. The integral ◮ The trick is to make the concepts and rules seem The difficulty of defining intuitively plausible functions ◮ by appealing to visual (geometric) intuition, or physical What the student takes intuition etc. away Calculus with limits: why teach it? Conclusions Appendix
What the student takes away Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Thus, apart from a bunch of rules, the student carries The difficulty of defining R away the following images: The difficulty of set theory function = graph The integral derivative = slope of tangent to graph The difficulty of defining functions integral = area under the curve. What the student takes away Calculus with limits: why teach it? Conclusions Appendix
What the student takes away Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Thus, apart from a bunch of rules, the student carries The difficulty of defining R away the following images: The difficulty of set theory function = graph The integral derivative = slope of tangent to graph The difficulty of defining functions integral = area under the curve. What the student takes ◮ The student is unable to relate the images to the rules. away Calculus with limits: why teach it? Conclusions Appendix
What the student takes away Calculus without Limits contd C. K. Raju Introduction The difficulty of limits ◮ Thus, apart from a bunch of rules, the student carries The difficulty of defining R away the following images: The difficulty of set theory function = graph The integral derivative = slope of tangent to graph The difficulty of defining functions integral = area under the curve. What the student takes ◮ The student is unable to relate the images to the rules. away Calculus with ◮ Ironically, the whole point of teaching limits is the belief limits: why teach it? that such visual intuition may be deceptive. Conclusions Appendix
Belief that visual intuition may deceive Calculus without Limits C. K. Raju ◮ Recall that Dedekind cuts were motivated by the doubt Introduction that the “fish figure” (Elements 1.1) is deceptive. The difficulty of N limits The difficulty of defining R The difficulty of set theory E W The integral The difficulty of defining functions What the student takes S away Calculus with Figure: The fish figure. limits: why teach it? Conclusions Figure: Dedekind’s doubt was that the two arcs which visually Appendix seem to intersect need not intersect since there are gaps in Q .
More misconceptions Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ Students in practice have more misconceptions. The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
More misconceptions Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ Students in practice have more misconceptions. The difficulty of ◮ They say the derivative is the slope of the tangent line defining R The difficulty of to a curve. set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
More misconceptions Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ Students in practice have more misconceptions. The difficulty of ◮ They say the derivative is the slope of the tangent line defining R The difficulty of to a curve. set theory ◮ And define a tangent as a line which meets the curve at The integral only one point. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
More misconceptions Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ Students in practice have more misconceptions. The difficulty of ◮ They say the derivative is the slope of the tangent line defining R The difficulty of to a curve. set theory ◮ And define a tangent as a line which meets the curve at The integral only one point. The difficulty of defining functions ◮ When pressed they see that a tangent line may meet a What the curve at more than one point. student takes away Calculus with limits: why teach it? Conclusions Appendix
More misconceptions Calculus without Limits C. K. Raju Introduction The difficulty of limits ◮ Students in practice have more misconceptions. The difficulty of ◮ They say the derivative is the slope of the tangent line defining R The difficulty of to a curve. set theory ◮ And define a tangent as a line which meets the curve at The integral only one point. The difficulty of defining functions ◮ When pressed they see that a tangent line may meet a What the curve at more than one point. student takes away ◮ But are unable to offer a different definition of the Calculus with tangent. limits: why teach it? Conclusions Appendix
Misconceptions about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ Such half-baked appeals to intuition confound the The difficulty of set theory student also from the perspective of physics. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Misconceptions about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ Such half-baked appeals to intuition confound the The difficulty of set theory student also from the perspective of physics. The integral ◮ In physical terms, the derivative is usually explained as The difficulty of “the rate of change”. defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Misconceptions about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R ◮ Such half-baked appeals to intuition confound the The difficulty of set theory student also from the perspective of physics. The integral ◮ In physical terms, the derivative is usually explained as The difficulty of “the rate of change”. defining functions ◮ But consider Popper’s argument. What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Popper’s argument about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of ◮ Velocity v = ∆ x ∆ t , then limits The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Popper’s argument about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of ◮ Velocity v = ∆ x ∆ t , then limits The difficulty of ◮ Choosing a large value of ∆ t will mean v is the average defining R velocity over the time period ∆ t , The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Popper’s argument about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of ◮ Velocity v = ∆ x ∆ t , then limits The difficulty of ◮ Choosing a large value of ∆ t will mean v is the average defining R velocity over the time period ∆ t , The difficulty of set theory ◮ this could differ substantially from the alertinstantaneous The integral velocity at any instant in that interval. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Popper’s argument about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of ◮ Velocity v = ∆ x ∆ t , then limits The difficulty of ◮ Choosing a large value of ∆ t will mean v is the average defining R velocity over the time period ∆ t , The difficulty of set theory ◮ this could differ substantially from the alertinstantaneous The integral velocity at any instant in that interval. The difficulty of defining ◮ But choosing small ∆ t will increase the relative error of functions measurement. What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Popper’s argument about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of ◮ Velocity v = ∆ x ∆ t , then limits The difficulty of ◮ Choosing a large value of ∆ t will mean v is the average defining R velocity over the time period ∆ t , The difficulty of set theory ◮ this could differ substantially from the alertinstantaneous The integral velocity at any instant in that interval. The difficulty of defining ◮ But choosing small ∆ t will increase the relative error of functions measurement. What the student takes away ◮ Hence there must be an optimum value of ∆ t neither Calculus with too large nor too small. limits: why teach it? Conclusions Appendix
Popper’s argument about rates of change Calculus without Limits C. K. Raju Introduction The difficulty of ◮ Velocity v = ∆ x ∆ t , then limits The difficulty of ◮ Choosing a large value of ∆ t will mean v is the average defining R velocity over the time period ∆ t , The difficulty of set theory ◮ this could differ substantially from the alertinstantaneous The integral velocity at any instant in that interval. The difficulty of defining ◮ But choosing small ∆ t will increase the relative error of functions measurement. What the student takes away ◮ Hence there must be an optimum value of ∆ t neither Calculus with too large nor too small. limits: why teach it? ◮ This is quite different from taking limits, and not at all Conclusions what calculus texts have in mind. Appendix
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory Calculus with limits: why teach it? The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits C. K. Raju ◮ If one is ultimately going to rely on (possibly faulty) Introduction The difficulty of visual and physical intuition limits The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits C. K. Raju ◮ If one is ultimately going to rely on (possibly faulty) Introduction The difficulty of visual and physical intuition limits ◮ why teach calculus with limits at all? The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits C. K. Raju ◮ If one is ultimately going to rely on (possibly faulty) Introduction The difficulty of visual and physical intuition limits ◮ why teach calculus with limits at all? The difficulty of defining R ◮ Why teach students to manipulate symbols they don’t The difficulty of set theory clearly understand? The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits C. K. Raju ◮ If one is ultimately going to rely on (possibly faulty) Introduction The difficulty of visual and physical intuition limits ◮ why teach calculus with limits at all? The difficulty of defining R ◮ Why teach students to manipulate symbols they don’t The difficulty of set theory clearly understand? The integral ◮ The human mind revolts at the thought of syntax devoid The difficulty of defining of semantics (as in the difficulty of assembly-language functions programming. What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits C. K. Raju ◮ If one is ultimately going to rely on (possibly faulty) Introduction The difficulty of visual and physical intuition limits ◮ why teach calculus with limits at all? The difficulty of defining R ◮ Why teach students to manipulate symbols they don’t The difficulty of set theory clearly understand? The integral ◮ The human mind revolts at the thought of syntax devoid The difficulty of defining of semantics (as in the difficulty of assembly-language functions programming. What the student takes ◮ This job of symbolic manipulation can be done more away easily by symbolic manipulation programs running on Calculus with limits: why teach low-cost computers. it? Conclusions Appendix
Why teach limits? Calculus without Limits C. K. Raju ◮ If one is ultimately going to rely on (possibly faulty) Introduction The difficulty of visual and physical intuition limits ◮ why teach calculus with limits at all? The difficulty of defining R ◮ Why teach students to manipulate symbols they don’t The difficulty of set theory clearly understand? The integral ◮ The human mind revolts at the thought of syntax devoid The difficulty of defining of semantics (as in the difficulty of assembly-language functions programming. What the student takes ◮ This job of symbolic manipulation can be done more away easily by symbolic manipulation programs running on Calculus with limits: why teach low-cost computers. it? ◮ Why teach human minds to think like low-cost Conclusions Appendix machines?
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, what the student learns in a calculus course defining R (manipulating unclearly defined symbols) The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, what the student learns in a calculus course defining R (manipulating unclearly defined symbols) The difficulty of set theory ◮ is a skill which has become obsolete. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, what the student learns in a calculus course defining R (manipulating unclearly defined symbols) The difficulty of set theory ◮ is a skill which has become obsolete. The integral ◮ it is today a useless skill. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, what the student learns in a calculus course defining R (manipulating unclearly defined symbols) The difficulty of set theory ◮ is a skill which has become obsolete. The integral ◮ it is today a useless skill. The difficulty of defining ◮ However, teaching the student to obey rules he does not functions What the understand student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of ◮ Thus, what the student learns in a calculus course defining R (manipulating unclearly defined symbols) The difficulty of set theory ◮ is a skill which has become obsolete. The integral ◮ it is today a useless skill. The difficulty of defining ◮ However, teaching the student to obey rules he does not functions What the understand student takes away ◮ teaches blind obedience to mathematical authority Calculus with (which lies in the West). limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory ◮ This sort of teaching started during colonialism. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Why teach limits? Calculus without Limits contd. C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory ◮ This sort of teaching started during colonialism. The integral The difficulty of ◮ But why should it continue today in a free society? defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Calculus without Limits C. K. Raju Introduction The difficulty of limits The difficulty of defining R The difficulty of set theory Conclusions The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R ◮ The definition of the derivative (which depends upon The difficulty of limits). set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R ◮ The definition of the derivative (which depends upon The difficulty of limits). set theory ◮ The definition of the limit (which depends upon R ). The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R ◮ The definition of the derivative (which depends upon The difficulty of limits). set theory ◮ The definition of the limit (which depends upon R ). The integral ◮ The definition of R (which depends upon set theory). The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R ◮ The definition of the derivative (which depends upon The difficulty of limits). set theory ◮ The definition of the limit (which depends upon R ). The integral ◮ The definition of R (which depends upon set theory). The difficulty of defining ◮ The definition of a set (which depends upon axiomatic functions set theory). What the student takes away Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R ◮ The definition of the derivative (which depends upon The difficulty of limits). set theory ◮ The definition of the limit (which depends upon R ). The integral ◮ The definition of R (which depends upon set theory). The difficulty of defining ◮ The definition of a set (which depends upon axiomatic functions set theory). What the ◮ The definition of the integral (which is defined only as student takes away an anti-derivative). Calculus with limits: why teach it? Conclusions Appendix
Conclusions Calculus without Limits C. K. Raju Introduction 1. The present-day course on calculus at the school (K-12) The difficulty of or beginning undergraduate level does NOT teach the limits following. The difficulty of defining R ◮ The definition of the derivative (which depends upon The difficulty of limits). set theory ◮ The definition of the limit (which depends upon R ). The integral ◮ The definition of R (which depends upon set theory). The difficulty of defining ◮ The definition of a set (which depends upon axiomatic functions set theory). What the ◮ The definition of the integral (which is defined only as student takes away an anti-derivative). Calculus with ◮ The definition of functions, such as e x . limits: why teach it? Conclusions Appendix
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