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Differential Equations

Differential Equations

Tangent Line Approximations (Again) Examples

Example 1

Let f (x) = 4x2 + 3.

(a) Find the tangent line to f at x = 3.
(b) Use the tangent line from part (a) to approximate f at 3.001.

Example 2

If f (1) = 2 and f ' (1) = 3, estimate f (1.2).

Example 3

The picture below shows the tangent line to the function f at x = 0. Estimate f (-0.1).

Example 4

The picture below shows a tangent line to f at x = a. What is f '(a)?

Example 5

Let f (x) = cos2 x. Use a tangent line approximation to estimate f (0.1).

Example 6

Let f (x) = x5 + 3x4 – 2x2x. Use a tangent line approximation to estimate f (0.9).

Example 7

Can a tangent line approximation ever produce the exact value of the function? Why or why not?

Example 8

The function y = f (x) is a solution to the IVP

Approximate f (1.01).