Exam 10: An Introduction to Calculus: Limits, Derivatives, and Integrals

$\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}$ (a) Explain why direct substitution cannot be used to find the limit (b) Find the limit algebraically, if it exists.

The following table lists the population statistics for a certain city. Year Population 1975 995,855 1985 990,928 1995 942,547 2005 889,370 What is the average rate of change in the population with respect to time between 1975 and 2005 ?

Use a calculator to find the LRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from x=0 to x=2 with 20 approximating rectangles.

Use a calculator to find the RRAM area approximation for the area under the graph $f(x)=-x^{3}+8$ from x=0 to x=2 with 20 approximating rectangles.

Write the integral that would be used to find the shaded area on the graph shown below.

At what points c in the domain of f(x) does $\lim _{x \rightarrow c} f(x)$ exist? $f(x)=\left\{\begin{array}{l}3 x & -5 \leq x<0 \\1 & x=0\\x^{2}+1& 0 < x \leq 2\\\end{array}\right.$

Find the derivative of f(x)=7 x+10 .

Use the graph below to find the limits or explain why the limits do not exist. (a) $\lim _{x \rightarrow 4^{-}} f(x)$ (b) $\lim _{x \rightarrow 4^{+}} f(x)$ (c) $\lim _{x \rightarrow 4} f(x)$

Draw the graph of $f(x)=-x(x-3)$ over the interval $[0,3]$ . On the graph, show and shade the rectangles that would be used to approximate the area under the curve $f(x)$ over $[0,3]$ by the right rectangle approximation method using 6 subintervals. Compute the estimation.

Determine $\lim _{x \rightarrow 0} \frac{\sin x}{x^{2}-x}$ if the limit exists.

Find the average rate of change of f(x)=5 \tan x over the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$

Write the integral that would be used to find the shaded area on the graph shown below.

Explain how to find the area under the graph of $f(x)=|x-2|$ from x=0 to x=4 by computing the geometric area.

Compute the integral $\int_{2}^{5} 4 x d x$ by computing a geometric area. (Hint: Graph the function first.)

Find the average rate of change of f(x)=3 \cos x over the interval $\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]$

What is the derivative of $f(x)=4 x^{2} ?$

Use the graph below to find the limits or explain why the limits do not exist. (a) $\lim _{x \rightarrow 2^{-}} f(x)$ (b) $\lim _{x \rightarrow 2^{+}} f(x)$ (c) $\lim _{x \rightarrow 2} f(x)$

The following table lists the population statistics for a certain city. Year Population 1975 695,854 1985 720,928 1995 742,547 2005 750,370 What is the average rate of change in the population with respect to time between 1975 and 2005 ?

Find the derivative of f(x)=-4 x-8 .

What is the average rate of change of $f(x)=\sqrt{3 x-1}$ over the interval [1.9,2.1] ?