Calculus Review Session Brian Prest Duke University Nicholas - PowerPoint PPT Presentation

Calculus Review Session Brian Prest Duke University Nicholas School of the Environment August 18, 2017

Topics to be covered 1. Functions and Continuity 2. Solving Systems of Equations 3. Derivatives (one variable) 4. Exponentials and Logarithms 5. Derivatives (multiple variables) 6. Integration 7. Optimization 2 ***Please ask questions at any time!

Functions and Continuity

Functions, continuous functions • Function: 𝑔(𝑦) – Mathematical relationship between variables ( input “x”, output “y”) – Each input (x) is related to one and only one output (y). – Easy graphical test: does an arbitrary vertical line intersect in more than one place? 4

Functions, continuous functions 1 𝑔𝑝𝑠 0 ≤ 𝑦 < 1 • Are these functions? 𝑧 = |𝑦| 𝑧 = 2 𝑔𝑝𝑠 1 ≤ 𝑦 < 3 3 𝑔𝑝𝑠 𝑦 ≥ 3 𝑧 = 0.5𝑦 + 2 Yes! Yes! Yes! 𝑦 2 + 𝑧 2 = 1 |𝑧| = 𝑦 𝑧 = 3 No! No! Yes! 5

Functions, continuous functions • Continuous : a function for which small changes in x result in small changes in y. – No holes, skips, or jumps – Intuitive test: can you draw the function without lifting your pen from the paper? 6

Functions, continuous functions 1 𝑔𝑝𝑠 0 ≤ 𝑦 < 1 • Are these continuous functions? 𝑧 = 2 𝑔𝑝𝑠 1 ≤ 𝑦 < 3 𝑧 = |𝑦| 3 𝑔𝑝𝑠 𝑦 ≥ 3 𝑧 = 0.5𝑦 + 2 Yes! No! Yes! 𝑦 2 + 𝑧 2 = 1 |𝑧| = 𝑦 No! (not a 𝑧 = 3 No! (not a function) function) Yes! 7

Solving Systems of Equations

Solving systems of linear equations • Count number of equations, number of unknowns – If # equations = # unknowns, unique solution might be possible Example: A. 𝑧 = 𝑦 B. 𝑧 = 2 − 𝑦 Answer : 𝑦 = 1, 𝑧 = 1 – If # equations < # unknowns, no unique solution. Example: • a = 1 − 2𝑐 . What is the value of b? Impossible. Any value works – If # equations > # unknowns, generally no solution that satisfies all of them. Example: • A and B above and third equation: 𝑧 = 1 + 𝑦 • Algebraic solutions • Graphical solution 9

Economics Example: Algebraic Approach • Economics Example 1. Supply: 𝑅 𝑡𝑣𝑞𝑞𝑚𝑗𝑓𝑒 = 2 ∗ 𝑄𝑠𝑗𝑑𝑓 2. Demand: 𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 = 4 − 2 ∗ 𝑄𝑠𝑗𝑑𝑓 3. “Market Clearing”: 𝑅 𝑡𝑣𝑞𝑞𝑚𝑗𝑓𝑒 = 𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 • Substitute equations (1) and (2) into (3) and solve for Price. – 2 ∗ 𝑄𝑠𝑗𝑑𝑓 = 4 − 2 ∗ 𝑄𝑠𝑗𝑑𝑓 – 4 ∗ 𝑄𝑠𝑗𝑑𝑓 = 4 → 𝑄𝑠𝑗𝑑𝑓 = 1 • Substitute price back into (1) and (2) – 𝑅 𝑡𝑣𝑞𝑞𝑚𝑗𝑓𝑒 = 2 ∗ 𝑄𝑠𝑗𝑑𝑓 = 2 – 𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 = 4 − 2 ∗ 𝑄𝑠𝑗𝑑𝑓 = 2 • 𝑸𝒔𝒋𝒅𝒇 = 𝟐 , 𝑹 𝒕𝒗𝒒𝒒𝒎𝒋𝒇𝒆 = 𝟑, 𝑹 𝒆𝒇𝒏𝒃𝒐𝒆𝒇𝒆 = 𝟑

Economics Example: Graphical Approach • Economics Example Supply: 𝑅 𝑡𝑣𝑞𝑞𝑚𝑗𝑓𝑒 = 2 ∗ 𝑄𝑠𝑗𝑑𝑓 1. Demand: 𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 = 4 − 2 ∗ 𝑄𝑠𝑗𝑑𝑓 2. “Market Clearing”: 𝑅 𝑡𝑣𝑞𝑞𝑚𝑗𝑓𝑒 = 𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 3. • Rearrange equations so they’re all in the same “y=” terms 1 2 𝑅 𝑡𝑣𝑞𝑞𝑚𝑗𝑓𝑒 – (1) becomes 𝑄𝑠𝑗𝑑𝑓 = −2 (𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 − 4) = 2 − 1 1 2 𝑅 𝑒𝑓𝑛𝑏𝑜𝑒𝑓𝑒 – (2) becomes 𝑄𝑠𝑗𝑑𝑓 = 2.5 2 1.5 Price Demand 1 Supply 0.5 0 0 1 2 3 4 5 Quantity

Derivatives

Differentiation, also known as taking the derivative (one variable) • The derivative of a function is the rate of change of the function. – Often denote it as: 𝑔 ′ 𝑦 or 𝑒 𝑒𝑔 𝑒𝑦 𝑔(𝑦) or 𝑒𝑦 – Be comfortable using these Δ𝑦 interchangeably Δ𝑔 • We can interpret the derivative (at a particular point) as the slope of the tangent line at that point. – If there is a small change in x, how much does f(x) change? 13

Differentiation, also known as taking the derivative (one variable) • Linear function: derivative is a constant, the slope 𝑒𝑧 (if 𝑧 = 𝑛𝑦 + 𝑐 , then 𝑒𝑦 = 𝑛 ) • Nonlinear function: derivative is not constant, but rather a function of x. Linear Function Non-Linear Function 𝑔 𝑦 = −𝑦 2 + 5𝑦 − 2 𝑔(𝑦) = 2𝑦 − 2 14

Differentiable functions • Differentiable: A function is differentiable at a point when there's a defined derivative at that point. – Algebraic test: if you know the equation and can solve for the derivative – Graphical test: “slope of tangent line of points from the left is approaching the same value as slope of the tangent line of the points from the right” – Intuitive graphical test: “As I zoom in, does the function tend to become a straight line ?” • Continuously differentiable function : differentiable everywhere x is defined. – That is, everywhere in the domain. If not stated, then negative to positive infinity. • If differentiable, then continuous. – However, a function can be continuous and not differentiable! (e.g., 𝑧 = |𝑦| ) • Why do we care? – When a function is differentiable, we can use all the power of calculus. 15

Functions, continuous functions 1 𝑔𝑝𝑠 0 ≤ 𝑦 < 1 • Are these continuously differentiable functions? 𝑧 = 2 𝑔𝑝𝑠 1 ≤ 𝑦 < 3 𝑧 = |𝑦| 3 𝑔𝑝𝑠 𝑦 ≥ 3 𝑧 = 0.5𝑦 + 2 No! No! Yes! 𝑦 2 + 𝑧 2 = 1 |𝑧| = 𝑦 No! (not a 𝑧 = 3 No! (not a function) function) Yes! 16

Rules of differentiation (one variable) • Power rule – 𝑧 = 𝑙𝑦 𝑏 → 𝑒𝑧 𝑒𝑦 = 𝑏𝑙𝑦 𝑏−1 – E.g., 𝑧 = 2𝑦 3 → 𝑒𝑧 𝑒𝑦 = 6𝑦 2 • Derivative of a constant – 𝑧 = 𝑙 → 𝑒𝑧 𝑒𝑦 = 0 𝑒𝑧 – E.g., 𝑧 = 3 → 𝑒𝑦 = 0 • Chain rule → 𝑒𝑧 𝑒𝑦 = 𝑒𝑔 𝑒𝑕 – 𝑧 = 𝑔 𝑕 𝑦 𝑒𝑕 𝑒𝑦 – E.g., 𝑧 = 1 + 7𝑦 2 → 𝑒𝑧 𝑒𝑦 = 2 1 + 7𝑦 ∗ 7 = 14 + 98𝑦 – 𝑔 𝑕(𝑦) = 𝑕(𝑦) 2 𝑕 𝑦 = 1 + 7𝑦 17

Rules of differentiation (one variable) • Addition rule – 𝑧 = 𝑔 𝑦 + 𝑕 𝑦 → 𝑒𝑧 𝑒𝑦 = 𝑒𝑔 𝑒𝑦 + 𝑒𝑕 𝑒𝑦 – E.g., 𝑧 = 2𝑦 + 𝑦 3 → 𝑒𝑧 𝑒𝑦 = 2 + 3𝑦 2 • Product rule – 𝑧 = 𝑔 𝑦 ∗ 𝑕 𝑦 → 𝑒𝑧 𝑒𝑦 = 𝑒𝑔 𝑒𝑦 𝑕 𝑦 + 𝑒𝑕 𝑒𝑦 𝑔 𝑦 – E.g., 𝑧 = 𝑦 2 3𝑦 + 1 → 𝑒𝑧 𝑒𝑦 = 2𝑦 3𝑦 + 1 + 3𝑦 2 = 9𝑦 2 + 2𝑦 • Quotient rule 𝑒𝑔 𝑒𝑦 𝑕 𝑦 − 𝑒𝑕 𝑒𝑦 𝑔 𝑦 – 𝑧 = 𝑔 𝑦 𝑕 𝑦 → 𝑒𝑧 𝑒𝑦 = 𝑕 𝑦 2 𝑦 2 𝑒𝑦 = 2𝑦 3𝑦+1 −3𝑦 2 3𝑦+1 → 𝑒𝑧 – E.g., 𝑧 = 3𝑦+1 2 18

Second, third and higher derivatives • Derivative of the derivative • Easy: Differentiate the function again (and again, and again …) • Some functions (polynomials without fractional or negative exponents) reduce to zero, eventually – 𝑧 = 𝑦 2 – First derivative = 2𝑦 – Second derivative = 2 – Third derivative = 0 • Other functions may not reduce to zero: e.g., 𝑔 𝑦 = 𝑓 𝑦 19

Exponentials and Logarithms

Exponents and logarithms (logs) • Logs are incredibly useful for understanding exponential growth and decay – half-life of radioactive materials in the environment – growth of a population in ecology – effect of discount rates on investment in energy-efficient lighting • Logs are the inverse of exponentials, just like addition:subtraction and multiplication:division 𝑧 = 𝑐 𝑦 ↔ log 𝑐 𝑧 = 𝑦 • In practice, we most often use base 𝑓 (Euler's number, 2.71828182846 …). – We write this as “ln”: ln 𝑦 = log 𝑓 𝑦 . • Sometimes, we also use base 10. • When in doubt, use natural log – Important! In Excel, LOG() is base 10 and LN() is natural log 21

Rules of logarithms • Logarithm of exponential function: ln 𝑓 𝑦 = 𝑦 – log of exponential function (more generally): ln 𝑓 𝑕 𝑦 = 𝑕(𝑦) • Exponential of log function: 𝑓 ln 𝑦 = 𝑦 – More generally, 𝑓 ln ℎ 𝑦 = ℎ(𝑦) • Log of products: ln 𝑦𝑧 = ln 𝑦 + ln 𝑧 𝑦 • Log of ratio or quotient: ln 𝑧 = ln 𝑦 − ln 𝑧 • Log of a power: ln 𝑦 𝑙 = 𝑙 ln 𝑦 – E.g., ln 𝑦 2 = 2ln(𝑦) 22

Derivatives of logarithms 𝑒 1 • 𝑒𝑦 ln 𝑦 = 𝑦 . This is just a rule. You have to memorize it. 𝑒 • What about 𝑒𝑦 ln 2𝑦 ? 𝑒 1 1 – Chain Rule: 𝑒𝑦 ln 2𝑦 = 2𝑦 2 = 𝑦 – Or, use the fact that ln 2𝑦 = ln 2 + ln 𝑦 and take the derivative of each term. (Simpler.) 𝑒 𝑒 1 – Also, this means 𝑒𝑦 ln 𝑙𝑦 = 𝑒𝑦 ln 𝑙 + ln 𝑦 = 𝑦 • … for any constant 𝑙 > 0 . • ( ln 𝐵 is defined only for 𝐵 > 0 .) 𝑒 • In general for 𝑒𝑦 ln 𝑕(𝑦) , where g(x) is any function of x, use the Chain Rule. 𝑒𝑦 • Why useful? Log-changes give percentages: 𝑒 ln 𝑦 = 𝑦 23