1. DO NOW
Find the equation of the tangent line of y = 3sin2x at x = ∏

2. 4.5 – Linearization and Newton’s Method
HW: 4.5 WKSH

4. Linearization
If f is differentiable at x = a, then the equation of the tangent line,
L(x) = f(a) + f’(a)(x – a)
Defines the linearization of f at a. The approximation f(x) ≈ L(x) is the standard linear approximation of f at a. The point x = a is the center of the approximation.

5. Example 1
Find the linearization of f(x) = √(1 + x) at x = 0, and use it to approximate √1.02 without a calculator.

6. Example 2
Find the linearization of f(x) = cosx at x = π/2 and use it to approximate cos(1.75) without a calculator.

7. Example 3
Use linearizations to approximate
√123
3√123

8. Differentials
Let y = f(x) be a differentiable function. The differential dx is an independent variable. The differential dy is
dy = f’(x) dx

9. Example 4
Find the differential dy and evaluate dy for the given values of x and dx.
y = x5 + 37x x = 1, dx = 0.01
y = sin3x x = π, dx = -0.02
x + y = xy x = 2, dx = 0.05

10. Example 5
(a) d(tan2x) =
(b)

11. Differential Estimate of Change
Let f(x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is
df = f’(a)dx.

12. Example 6
The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in the circle’s area A. Compare this estimate with the true change ∆A, and find the approximation error.

13. Absolute, Relative, and Percentage Change
True
Estimate
Absolute Change
∆f = f(a + dx) – f(a)
df = f’(a)dx
Relative Change
∆f/f(a)
df/f(a)
Percentage Change
∆f/f(a) x 100
df/f(a) x 100

14. Example 7
Inflating a bicycle tire changes its radius from 12 inches to 13 inches. Use differentials to estimate the absolute change, the relative change, and the percentage change in the perimeter of the tire.