MA 3972-MA-Book May 9, 2018 10:9112 STEP 4. Review the Knowledge You Need to Score High
Example 2
Evaluate limh→ 0
cos(π+h)−cos(π)
h.
The expression lim
h→ 0cos(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
=hlim→ 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 7.1-6.)[–3.14, 6.28] by [–3, 3]
Figure 7.1-6Example 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 7.1-7.)[–5, 5] by [–1, 6]
Figure 7.1-7Example 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.