# 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*) = *x*^{3} + 2*x* + 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*) = log_{2} *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*) = 4*x* + *e*^{x}, *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*) = 3*x*^{2} - 4*x* + 1. Use a tangent line approximation to estimate *f* (2.01).

### Example 12

Let *f* (*x*) = 7*x*^{2} - 3*x* + 4. Use a tangent line approximation to estimate *f* (2.99).

### Example 13

Let *f* (*x*) = *e*^{x}. Use a tangent line approximation to estimate *f* (0.05).

### Example 14

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

### Example 15

Let *f* (*x*) = sin (5*x*). 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).