5 Steps to a 5 AP Calculus BC 2019

78 STEP 4. Review the Knowledge You Need to Score High

Example 2

Evaluate limh→ 0

cos(π+h)−cos(π)

h

.

The expression limh→ 0

cos(π+h)−cos(π)

h

is equivalent to the derivative of the function

f(x)=cosxatx=π, i.e., f′(π). The derivative of f(x)=cosxatx=πis equivalent to

the slope of the tangent to the curve of cosxatx=π. The tangent is parallel to thex-axis.

Thus, the slope is 0 or limh→ 0

cos(π+h)−cos(π)

h

= 0.

Or, using an algebraic method, note that cos(a+b)=cos(a) cos(b)−sin(a) sin(b).

Then rewrite limh→ 0

cos(π+h)−cos(π)

h

=limh→ 0

cos(π) cos(h)−sin(π) sin(h)−cos(π)

h

=

limh→ 0

−cos(h)−(−1)

h

=limh→ 0

−cos(h)+ 1

h

=limh→ 0

−[cos(h)−1]

h

=−limh→ 0

[cos(h)−1]

h

= 0.

(See Figure 6.1-6.)

[−3.14,6.28] by [−3,3]

Figure 6.1-6

Example 3

If the functionf(x)=x^2 /^3 +1, find all points where fis not differentiable.

The function f(x) is continuous for all real numbers and the graph of f(x) forms a

“cusp” at the point (0, 1). Thus,f(x) is not differentiable atx=0. (See Figure 6.1-7.)

[−5,5] by [−1,6]

Figure 6.1-7

Example 4

Using a calculator, find the derivative off(x)=x^2 + 4 xatx=3.

There are several ways to findf′(3), using a calculator. One way is to use the [nDeriv]

function of the calculator. From the main Home screen, selectF3-Calcand then select

[nDeriv]. Enter [nDeriv](x^2 + 4 x, x)|x=3. The result is 10.