Chapter 1
Calculus Early Transcendentals: Pearson New International Edition · 441 exercises
Problem 26
Find each value without using a calculator $$ \tan \left[2 \tan ^{-1}\left(\frac{1}{3}\right)\right] $$
6 step solution
Problem 26
Which of the following are odd functions? Even functions? Neither? (a) \(\cot t+\sin t\) (b) \(\sin ^{3} t\) (c) \(\sec t\) (d) \(\sqrt{\sin ^{4} t}\) (e) \(\cos (\sin t)\) (f) \(x^{2}+\sin x\)
7 step solution
Problem 26
Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ F(t)=-|t+3| $$
4 step solution
Problem 26
, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=x^{2}(x-1)(x-2) $$
5 step solution
Problem 26
$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{2 x-2 x^{2}}{x^{3}-2 x^{2}+x} $$
4 step solution
Problem 27
In Problems \(23-28\), find the slope of the line containing the given two points. (3,0) \text { and }(0,5)
4 step solution
Problem 27
Find each value without using a calculator $$ \sin \left[\cos ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{5}{13}\right)\right] $$
6 step solution
Problem 27
Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(x)=\left\lfloor\frac{x}{2}\right\rfloor $$
6 step solution
Problem 27
, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=x^{2}(x-1)^{2} $$
5 step solution
Problem 27
Tell whether each of the following is true or false. (a) \(-3<-7\) (b) \(-1>-17\) (c) \(-3<-\frac{22}{7}\)
3 step solution
Problem 27
$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{12}{x^{2}+2 x}+\frac{4}{x}+\frac{2}{x+2} $$
5 step solution
Problem 28
In Problems \(23-28\), find the slope of the line containing the given two points. (-6,0) \text { and }(0,6)
4 step solution
Problem 28
Find each value without using a calculator $$ \cos \left[\cos ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{12}{13}\right)\right] $$
5 step solution
Problem 28
Find the exact values in Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$
4 step solution
Problem 28
Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ G(x)=[2 x-1] $$
4 step solution
Problem 28
, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=x^{4}(x-1)^{4}(x+1)^{4} $$
5 step solution
Problem 28
Tell whether each of the following is true or false. (a) \(-5>-\sqrt{26}\) (b) \(\frac{6}{7}<\frac{34}{39}\) (c) \(-\frac{5}{7}<-\frac{44}{59}\)
3 step solution
Problem 28
$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{2}{6 y-2}+\frac{y}{9 y^{2}-1} $$
7 step solution
Problem 29
In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) \text { Through }(2,2) \text { with slope }-1
5 step solution
Problem 29
Show that each equation is an identity. $$ \tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^{2}}} $$
3 step solution
Problem 29
Find the exact values in Hint: Half-angle identities may be helpful. $$ \sin ^{3} \frac{\pi}{6} $$
3 step solution
Problem 29
Specify whether the given function is even, odd, or neither, and then sketch
its graph.
$$
g(t)=\left\\{\begin{array}{ll}
1 & \text { if } t \leq 0 \\
t+1 & \text { if } 0
5 step solution
Problem 29
, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ |x|+|y|=1 $$
6 step solution
Problem 29
Assume that \(a>0, b>0\). Prove each statement. Hint: Each part requires two proofs: one for \(\Rightarrow\) and one for \(\Leftarrow .\) (a) \(a\frac{1}{b}\)
4 step solution
Problem 29
. Find the value of each of the following; if undefined, say so. (a) \(0 \cdot 0\) (b) \(\frac{0}{0}\) (c) \(\frac{0}{17}\) (d) \(\frac{3}{0}\) (e) \(0^{5}\) (f) \(17^{0}\)
6 step solution
Problem 30
In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) \text { Through }(3,4) \text { with slope }-1
4 step solution
Problem 30
Show that each equation is an identity. $$ \sin \left(\tan ^{-1} x\right)=\frac{x}{\sqrt{1+x^{2}}} $$
4 step solution
Problem 30
Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ h(x)=\left\\{\begin{array}{ll} -x^{2}+4 & \text { if } x \leq 1 \\ 3 x & \text { if } x>1 \end{array}\right. $$
5 step solution
Problem 30
, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ |x|+|y|=4 $$
4 step solution
Problem 30
Which of the following are true if \(a \leq b ?\) (a) \(a^{2} \leq a b\) (b) \(a-3 \leq b-3\) (c) \(a^{3} \leq a^{2} b\) (d) \(-a \leq-b\)
4 step solution
Problem 30
Show that division by 0 is meaningless as follows: Suppose that \(a \neq 0\). If \(a / 0=b\), then \(a=0 \cdot b=0\), which is a contradiction. Now find a reason why \(0 / 0\) is also meaningless.
2 step solution
Problem 31
In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) With \(y\) -intercept 3 and slope 2
3 step solution
Problem 31
How are \(\log _{1 / 2} x\) and \(\log _{2} x\) related?
5 step solution
Problem 31
In Problems \(31-44\), find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=x+1 $$
4 step solution
Problem 31
Show that each equation is an identity. $$ \cos \left(2 \sin ^{-1} x\right)=1-2 x^{2} $$
5 step solution
Problem 31
Find the exact values in Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{8} $$
5 step solution
Problem 31
A plant has the capacity to produce from 0 to \(100 \mathrm{com}\) puters per day. The daily overhead for the plant is \(\$ 5000\), and the direct cost (labor and materials) of producing one computer is \(\$ 805 .\) Write a formula for \(T(x)\), the total cost of producing \(x\) computers in one day, and also for the unit cost \(u(x)\) (average cost per computer). What are the domains of these functions?
4 step solution
Problem 31
31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs, $$ \begin{array}{l} y=-x+1 \\ y=(x+1)^{2} \end{array} $$
6 step solution
Problem 31
Find all values of \(x\) that satisfy both inequalities simultaneously. (a) \(3 x+7>1\) and \(2 x+1<3\) (b) \(3 x+7>1\) and \(2 x+1>-4\) (c) \(3 x+7>1\) and \(2 x+1<-4\)
9 step solution
Problem 31
31-36, change each rational number to a decimal by performing long division. $$ \frac{1}{12} $$
8 step solution
Problem 32
In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) With \(y\) -intercept 5 and slope 0
4 step solution
Problem 32
Sketch the graphs of \(\log _{1 / 3} x\) and \(\log _{3} x\) using the same coordinate axes.
7 step solution
Problem 32
Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=-\frac{x}{3}+1 $$
5 step solution
Problem 32
Show that each equation is an identity. $$ \tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}} $$
5 step solution
Problem 32
Find identities analogous to the addition identities for each expression. (a) \(\sin (x-y)\) (b) \(\cos (x-y)\) (c) \(\tan (x-y)\)
6 step solution
Problem 32
It costs the \(\mathrm{ABC}\) Company \(400+5 \sqrt{x(x-4)}\) dollars to make \(x(x \geq 4)\) toy stoves that sell for \(\$ 6\) each. (a) Find a formula for \(P(x)\), the total profit in making \(x\) stoves. (b) Evaluate \(P(200)\) and \(P(1000)\). (c) How many stoves does \(\mathrm{ABC}\) have to make to just break even?
5 step solution
Problem 32
plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs. $$ \begin{array}{l} y=2 x+3 \\ y=-(x-1)^{2} \end{array} $$
6 step solution
Problem 32
Find all the values of \(x\) that satisfy at least one of the two inequalities. (a) \(2 x-7>1\) or \(2 x+1<3\) (b) \(2 x-7 \leq 1\) or \(2 x+1<3\) (c) \(2 x-7 \leq 1\) or \(2 x+1>3\)
9 step solution
Problem 32
change each rational number to a decimal by performing long division. $$ \frac{2}{7} $$
6 step solution
Problem 33
In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) Through \((2,3)\) and \((4,8)\)a
4 step solution