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Differential Equations

Differential Equations

Example 1

For the function,

(a) Find the tangent line to the function at the specified value of x.

(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).

f (x) = x3 + 2x + 1, x = 1, Δ x = .2

Example 2

For the function,

(a) Find the tangent line to the function at the specified value of x.

(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).

Example 3

For the function,

(a) Find the tangent line to the function at the specified value of x.

(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).

f (x) = log2 x, x = 8, Δ x = .01

Example 4

For the function,

(a) Find the tangent line to the function at the specified value of x.

(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).

f (x) = 4x + ex, x = 0, Δ x = 0.05

Example 5

For the function,

(a) Find the tangent line to the function at the specified value of x.

(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).

Example 6

If f (3) = 4.2 and f '(3) = 0.15, estimate f (3.1).

Example 7

If and , estimate .

Example 8

The picture below shows the tangent line to f at -1. Estimate f (-.9).

Example 9

The picture below shows the tangent line to f at x = 3. Find f '(3).

Example 10

If f (2) = 7 and a tangent line approximation at x = 2 estimates f (2.1) to be 7.2, what is f '(2)?

Example 11

Let f (x) = 3x2 - 4x + 1. Use a tangent line approximation to estimate f (2.01).

Example 12

Let f (x) = 7x2 - 3x + 4. Use a tangent line approximation to estimate f (2.99).

Example 13

Let f (x) = ex. Use a tangent line approximation to estimate f (0.05).

Example 14

Let f (x) = 2x. Use a tangent line approximation to estimate f (3.02).

Example 15

Let f (x) = sin (5x). Use a tangent line approximation to estimate f (3).

Example 16

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

Estimate f (0.5).

Example 17

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

Estimate f (-½).

Example 18

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

Estimate f (2.01).

Example 19

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

Approximate f (6.95).

Example 20

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

Approximate f (3.01).

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