91Ó°ÊÓ

\(G(x)=-x^{4}+32 x^{2}+144\) (a) Determine whether \(G\) is even, odd, or neither. (b) There is a local maximum value of 400 at \(x=4\). Find a second local maximum value. (c) Suppose the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=0\) and \(x=6\) is 1612.8 square units. Using the result from (a), determine the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=-6\) and \(x=0\).

Ethan has $$\$ 60,000$$ to invest. He puts part of the money in a CD that earns \(3 \%\) simple interest per year and the rest in a mutual fund that earns \(8 \%\) simple interest per year. How much did he invest in each if his earned interest the first year was $$\$ 3700$$.

Suppose (1,3) is a point on the graph of \(y=f(x)\) (a) What point is on the graph of \(y=f(x+3)-5 ?\) (b) What point is on the graph of \(y=-2 f(x-2)+1 ?\) (c) What point is on the graph of \(y=f(2 x+3) ?\)

Find the quotient and remainder when \(x^{3}+3 x^{2}-6\) is divided by \(x+2\)

Given \(f(x)=\frac{1}{x}\) and \(\left(\frac{f}{g}\right)(x)=\frac{x+1}{x^{2}-x},\) find the function \(g\)

For the function \(f(x)=x^{2},\) compute the average rate of change: \(\begin{array}{ll}\text { (a) From } 1 \text { to } 2 & \text { (b) From } 1 \text { to } 1.5\end{array}\) (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with \(f\) (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=x^{2}+2 x\)

The period \(T\) (in seconds) of a simple pendulum is a function of its length \(l\) (in feet) defined by the equation $$ T=2 \pi \sqrt{\frac{l}{g}} $$ where \(g \approx 32.2\) feet per second per second is the acceleration due to gravity. (a) Use a graphing utility to graph the function \(T=T(l)\). (b) Now graph the functions \(T=T(l+1), T=T(l+2)\) $$ \text { and } T=T(l+3) $$ (c) Discuss how adding to the length \(l\) changes the period \(T\) (d) Now graph the functions \(T=T(2 l), T=T(3 l)\), and \(T=T(4 l)\) (e) Discuss how multiplying the length \(l\) by factors of 2,3 , and 4 changes the period \(T\)

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=2 x^{2}-3 x+1\)

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=-x^{2}+3 x-2\)