- Topics At a Glance
- Differential Equations and Their Solutions
- Solutions to Differential Equations
- Solving Differential Equations
- Initial Value Problems
- More About Solutions
- Word Problems
- Slope Fields
- Slope Fields and Solutions
- Equilibrium Solutions
**Euler's Method**- Slopes (Again)
**Tangent Line Approximations (Again)**- The Scoop on Euler
- Accuracy and Usefulness of Euler's Method
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Tangent line approximation can also be called **local linearization**, **linear approximation**, and probably a bunch of other names. The important thing is that you're *using a line to approximate a curve*.

Here's a reminder of how tangent line approximation works:

Suppose we know the value of *f* at a particular value of *x*:

and we want to know the value of *f* at a nearby value of *x*:

We draw the tangent line to *f* at the point we know:

Then we find the value on the tangent line at the nearby value of *x*:

This tangent line value is close to the value we actually wanted.

Using the language from the last section, let's talk this through again.

We know y_{old}:

We want to know the value of *f* if we change *x* by Δ *x*:

We draw the tangent line to *f* at the point we know:

and we use the tangent line to find y_{new}, which is close to the value we actually wanted.

Another kind of problem you may run into is the kind where you're given a formula for the function *f* and asked to use a tangent line to approximate *f* at some particular value *x*^{*}.

To do this, pick some *x* = a close to *x*^{*}. Choose a so that *f* (*a*) and *f* '(*a*) are easy to calculate. Then we have y_{old} = *f* (*a*), Δ *x* = *x*^{*} - *a*, and slope = *f* '(*a*).

From there you know how to calculate y_{new}, which is the approximation you want.

Example 1

Let (a) Find the tangent line to |

Example 2

If |

Example 3

The picture below shows the tangent line to the function |

Example 4

The picture below shows a tangent line to |

Example 5

Let |

Example 6

Let |

Example 7

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

Example 8

The function Approximate |

Exercise 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

Exercise 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*).

Exercise 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

Exercise 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

Exercise 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*).

Exercise 6

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

Exercise 7

If and , estimate .

Exercise 8

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

Exercise 9

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

Exercise 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)?

Exercise 11

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

Exercise 12

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

Exercise 13

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

Exercise 14

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

Exercise 15

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

Exercise 16

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

Estimate *f* (0.5).

Exercise 17

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

Estimate *f* (-½).

Exercise 18

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

Estimate *f* (2.01).

Exercise 19

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

Approximate *f* (6.95).

Exercise 20

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

Approximate *f* (3.01).