Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 10

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

$h\left(x\right)=-4x^{{}^2}-4x+12$

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 4x-3

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = ƒ⁻¹ (h)

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

d. At what time is the ball at a height of 30 ft on the way up?

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

e. At what time is the ball at a height of 10 ft on the way down? 

Find a trigonometric function $f$ represented by the graph in the figure. <IMAGE>

Prove the following identities.

$\sec\theta=\frac{1}{\cos\theta}$

Prove the following identities.

$\tan\theta=\frac{\sin\theta}{\cos\theta}$

Changing bases Convert the following expressions to the indicated base.

$\ln \mid x\mid$ using base 5

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\tan^{-1}\sqrt3$

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Find the slope of the secant line that passes through points $A$ and $B$. Interpret your answer as an average rate of change over the interval $1\leq{t}\leq{3}$.

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Repeat the procedure outlined in part (a) for the secant line that passes through points $P$ and $Q$.

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Notice that the curve in the figure is horizontal for an interval of time near $t=5.5$ hr. Give a plausible explanation for the horizontal line segment.

Roots and powers Sketch a graph of the given pairs of functions. Be sure to draw the graphs accurately relative to each other.

y = (x)¹⸍³ and y = (x)¹⸍⁵

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points $P$ and $Q$ on the graph of the function.

a. Sketch a graph of the function and the secant line through $P$ and $Q$.

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After $t$ seconds, an object dropped from rest falls a distance $d=16t^2$, where $d$ is measured in feet and $2\leq{t}\leq{5}$.

Determine whether the following statements are true and give an explanation or counterexample.

$\frac{{\log }_{b}x}{{\log }_{b}y}={\log }_{b}x-{\log }_{b}y$

Determine whether the following statements are true and give an explanation or counterexample.

${\log }_5{4}^6=4{\log }_{5}6$

Determine whether the following statements are true and give an explanation or counterexample.

$2=10^{{\log }_{10}2}$

Determine whether the following statements are true and give an explanation or counterexample.

$2=\ln 2^e$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=x^2+1$ , then $f^{-1}\left(x\right)=\frac{1}{x^2+1}$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=\frac{1}{x}$ , then $f^{-1}\left(x\right)=\frac{1}{x}$

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\csc^{-1}\left(-1\right)$

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\tan^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right)$

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\tan^{-1}\left(\tan\left(\frac{3\pi}{4}\right)\right)$

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x{}^5-x^3-2$

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$\tan\left(\tan^{-1}1\right)$

Composition of polynomials

Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.

What is the degree of the following polynomials?

ƒ ⋅ f