Practice problems for these concepts can be found at: More Applications of Derivatives Practice Problems for AP Calculus

### Tangent Line Approximation (or Linear Approximation)

An equation of the tangent line to a curve at the point (*a*, *f* (*a*)) is:

*y* = *f'*(*a*)+ *f'*(*a*)(*x* – *a*) providing that *f* is differentiable at *a*. See Figure 9.2-1.

Since the curve of *f* (*x* ) and the tangent line are close to each other for points near *x* = *a*, *f* (*x* ) ≈ *f* (*a*)+ *f'*(*a*)(*x* – *a*).

#### Example 1

Write an equation of the tangent line to *f* (*x* ) = *x*^{3} at (2, 8). Use the tangent line to find the approximate values of *f* (1.9) and *f* (2.01).

Differentiate *f* (*x* ): *f'*(x )=3*x*^{2}; *f'*(2)=3(2)^{2} =12. Since *f* is differentiable at *x* =2, an equation of the tangent at *x* =2 is:

*y*=

*f*(2)+

*f'*(2)(

*x*– 2)

*y*=(2)

^{3}+12(

*x*– 2)=8+12

*x*– 24=12

*x*– 16

*f*(1.9) ≈ 12(1.9) – 16 = 6.8

*f*(2.01) ≈ 12(2.01) – 16 = 8.12. (See Figure 9.2-2.)

#### Example 2

If *f* is a differentiable function and *f* (2)=6 and , find the approximate value of f (2.1).

Using tangent line approximation, you have

*f*(2) = 6 the point of tangency is (2, 6);- the slope of the tangent at x =2 is
- the equation of the tangent is ;
- thus, .

#### Example 3

The slope of a function at any point (x, y) is . The point (3, 2) is on the graph of *f*. (a) Write an equation of the line tangent to the graph of *f* at *x* =3. (b) Use the tangent line in part (a) to approximate *f* (3.1).

- Let
*y*=*f*(*x*), then *f*(3.1) ≈ –2(3.1)+8 ≈ 1.8

.

Equation of tangent: *y* – 2= – 2(*x* – 3) or *y* = – 2*x* +8.

### Estimating the nth Root of a Number

Another way of expressing the tangent line approximation is:

*f*(*a* + Δ*x* ≈ *f*(*a*) + *f* '(*a* Δ *x*; where Δ *x* is a relatively small value.

#### Example 1

Find the approximation value of using linear approximation.

Using *f*(*a* + Δ*x* ≈ *f*(*a*) + *f* '(*a* Δ *x*, let Δ *x* = 1.

Thus, .

#### Example 2

Find the approximate value of using linear approximation.

Let , *a* = 64, Δ*x* = – 2. Since and , you can use *f*(*a* + Δ*x* ≈ *f*(*a*) + *f* '(*a* Δ *x*. Thus, .

### Estimating the Value of a Trigonometric Function of an Angle

### Example

Approximate the value of sin 31°.

Note: You must express the angle measurement in radians before applying linear approximations. 30° = radians and 1° = radians.

Let *f (x)* = sin *x*, *a* = and Δ*x* = .

Practice problems for these concepts can be found at:

More Applications of Derivatives Practice Problems for AP Calculus

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