Review Exercises
Finding Limits: A Numerical and Graphical Approach
For the following exercises, use Figure 1.
limx→−1−f(x)
limx→3f(x)
Using Table 1, estimate limx→0f(x).
x | F(x) |
−0.1 | 2.875 |
−0.01 | 2.92 |
−0.001 | 2.998 |
0 | Undefined |
0.001 | 2.9987 |
0.01 | 2.865 |
0.1 | 2.78145 |
0.15 | 2.678 |
For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a. If the function has limit as x approaches a, state it. If not, discuss why there is no limit.
f(x)={1x+1,ifx=−2(x+1)2,ifx≠−2 a=−2
Finding Limits: Properties of Limits
For the following exercises, find the limits if limx→cf(x)=−3 and limx→cg(x)=5.
limx→c(f(x)+g(x))
limx→c(f(x)⋅g(x))
limx→0−f(x),f(x)={3x2+2x+15x+3 x>0x<0
For the following exercises, evaluate the limits using algebraic techniques.
limh→0((h+6)2−36h)
limx→1(−x2−9xx)
limx→−3(13+1x3+x)
Continuity
For the following exercises, use numerical evidence to determine whether the limit exists at x=a. If not, describe the behavior of the graph of the function at x=a.
f(x)=−2(x−4)2;a=4
f(x)=6x2+23x+204x2−25;a=−52
For the following exercises, determine where the given function f(x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.
f(x)=x2−2x−15
f(x)=x2−2xx2−4x+4
f(x)=x2−1x2−x
f(x)=x+2x3+8
Derivatives
For the following exercises, find the average rate of change f(x+h)−f(x)h.
f(x)=5
f(x)=ln(x)
For the following exercises, find the derivative of the function.
f(x)=4x−6
Find the equation of the tangent line to the graph of f(x) at the indicated x value.
f(x)=−x3+4x; x=2.
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.
Given that the volume of a right circular cone is V=13πr2h and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of π