Todays Agenda Upcoming Homework Section 2.8: Linear Approximations - - PowerPoint PPT Presentation

Today’s Agenda

• Upcoming Homework
• Section 2.8: Linear Approximations and Differentials

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 1 / 8

Upcoming Homework

• WeBWorK HW #11 (Section 2.8), due 10/5
• Written HW F (Sections 2.7 and 2.8) due 10/7
• WeBWorK HW #12 (Sections 3.1 and 3.2) due 10/9

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 2 / 8

Section 2.8

Definition 2.8.1

Given a function f (x) that is differentiable at a point a, we say that the linear approximation of f at a, or the tangent line approximation of f at a, denoted L(x), is the function whose graph is the tangent line at a: L(x) = f (a) + f ′(a)(x − a).

Example 2.8.2

Find the linear approximation of f (x) = x4 + 3x2 at the point a = −1.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 3 / 8

Section 2.8

Linear approximations are helpful when you have a ”difficult” function to evaluate and a certain margin of error in computation is allowable. An almost trivial example follows:

Example 2.8.3

Let f (x) = tan x.

1 Find the linear approximation of f (x) at a = 0. 2 If an error in calculation of ±0.1 is allowable, find the values of x for

which L(x) can be used to approximate f (x). One example of when this is useful: the classic pendulum experiment in an introductory physics class. The tangential acceleration of the bob of the pendulum is given by the equation aT = −g sin θ, but for very small angles, sin θ ≈ θ, and so one can simply use aT = −9.8θ in the experiment instead of needing to calculate the sine for each piece of data.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 4 / 8

Section 2.8

Given a function f (x) and a change in x denoted by ∆x, recall that we can calculate the corresponding ∆y via the formula: ∆y = f (x + ∆x) − f (x). If we consider instead L(x), we can approximate the change in y by the formula dy = f ′(x) dx.

Definition 2.8.4

If y = f (x), where f is a differentiable function, then the differential dx is an independent variable; that is, dx can be given the value of any real

• number. The differential dy is then defined in terms of dx by the equation

dy = f ′(x) dx.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 5 / 8

Section 2.8

The formula for the differential can be a bit confusing, because the differential dy is specific to a point a. As an example, consider f (x) = tan x again. The differential at the point a = 0 is dy = sec2(0) dx = dx, but the differential at the point a = π/3 is dy = sec2(π/3) dx = 4 dx. We must be very careful to pay attention to whether a formula for the differential is required, or whether the differential at a specific point is required. Another confusing part of the differential is that its notation is very, very similar to the notation for the derivative dy/dx. It is tempting to interpret dy = dy dx dx as multiplying a fraction by its denominator and ”clearing the denominator.” However, remember that dy/dx is NOT a fraction; d dx y = dy dx is the action of the derivative operator on the function y.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 6 / 8

Section 2.8

To visualize ∆x, ∆y, dx, and dy, consider the following graph (found on page 137 of your textbook):

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 7 / 8

Section 2.8

Example Problems

1 The edge of a cube is measured to be 30 cm ±0.1 cm. 1

Use differentials to estimate the maximum possible error in computing the volume and surface area of the cube.

2

What is the volume of a 29.9 cm cube? How does this compare to your estimate in part (a)?

2 The radius of a circular disk is given as 24 cm with a maximum error

in measurement of 0.5 cm.

1

Use differentials to estimate the maximum error in calculating the area

• f the disk.

2

What is the actual area of a 24.5 cm disk? How does this compare with your estimate in part (a)?

3 Use differentials to estimate the amount of paint needed to apply a

coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m (recall that the volume of a sphere is given by V = 4 3πr3).

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 2 October 2015 8 / 8