Calculus: A Complete Course

Robert Adams , Christopher Essex

Chapter 2

Differentiation - all with Video Answers

Educators

Section 1

Tangent Lines and Their Slopes

Problem 1

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=3 x-1 \text { at }(1,2)$$

Sanchit Jain

Sanchit Jain

Numerade Educator

Problem 2

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=x / 2 \text { at }(a, a / 2)$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 3

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=2 x^{2}-5 \text { at }(2,3)$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 4

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=6-x-x^{2} \text { at } x=-2$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 5

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=x^{3}+8 \text { at } x=-2$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 6

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=\frac{1}{x^{2}+1} \text { at }(0,1)$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 7

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=\sqrt{x+1} \text { at } x=3$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 8

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=\frac{1}{\sqrt{x}} \text { at } x=9$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 9

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=\frac{2 x}{x+2} \text { at } x=2$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 10

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=\sqrt{5-x^{2}} \text { at } x=1$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 11

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=x^{2} \text { at } x=x_{0}$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 12

Find an equation of the straight line tangent to the given curve at the point indicated.
$$y=\frac{1}{x} \text { at }\left(a, \frac{1}{a}\right)$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 13

Do the graphs of the functions $f$ have tangent lines at the given points? If yes, what is the tangent line?
$$f(x)=\sqrt{|x|} \text { at } x=0$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 14

Do the graphs of the functions $f$ have tangent lines at the given points? If yes, what is the tangent line?
$$f(x)=(x-1)^{4 / 3} \text { at } x=1$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 15

Do the graphs of the functions $f$ have tangent lines at the given points? If yes, what is the tangent line?
$$f(x)=(x+2)^{3 / 5} \text { at } x=-2$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 16

Do the graphs of the functions $f$ have tangent lines at the given points? If yes, what is the tangent line?
$$f(x)=\left|x^{2}-1\right| \text { at } x=1$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 17

Do the graphs of the functions $f$ have tangent lines at the given points? If yes, what is the tangent line?
$$f(x)=\left\{\begin{array}{ll}
\sqrt{x} & \text { if } x \geq 0 \\
-\sqrt{-x} & \text { if } x<0
\end{array} \text { at } x=0\right.$$

Yujie Wang

College of San Mateo

Problem 18

Find the slope of the curve $y=x^{2}-1$ at the point $x=x_{0}$ What is the equation of the tangent line to $y=x^{2}-1$ that has slope $-3 ?$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 19

(a) Find the slope of $y=x^{3}$ at the point $x=a$
(b) Find the equations of the straight lines having slope 3 that are tangent to $y=x^{3}$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 20

Find all points on the curve $y=x^{3}-3 x$ where the tangent line is parallel to the $x$ -axis.

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 21

Find all points on the curve $y=x^{3}-x+1$ where the tangent line is parallel to the line $y=2 x+5$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 22

Find all points on the curve $y=1 / x$ where the tangent line is perpendicular to the line $y=4 x-3$

Willis James

Numerade Educator

Problem 23

For what value of the constant $k$ is the line $x+y=k$ normal to the curve $y=x^{2} ?$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 24

For what value of the constant $k$ do the curves $y=k x^{2}$ and $y=k(x-2)^{2}$ intersect at right angles? Hint: Where do the curves intersect? What are their slopes there?

Yujie Wang

College of San Mateo

Problem 25

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=x^{3}(5-x)^{2}$$

Yujie Wang

College of San Mateo

Problem 26

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=2 x^{3}-3 x^{2}-12 x+1$$

Yujie Wang

College of San Mateo

Problem 27

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=\left|x^{2}-1\right|-x$$

Yujie Wang

College of San Mateo

Problem 28

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=|x+1|-|x-1|$$

Yujie Wang

College of San Mateo

Problem 29

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=\left(x^{2}-1\right)^{1 / 3}$$

Yujie Wang

College of San Mateo

Problem 30

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=\left(\left(x^{2}-1\right)^{2}\right)^{1 / 3}$$

Yujie Wang

College of San Mateo

Problem 31

If line $L$ is tangent to curve $C$ at point $P$, then the smaller angle between $L$ and the secant line $P Q$ joining $P$ to another point $Q$ on $C$ approaches 0 as $Q$ approaches $P$ along $C$. Is the converse true: if the angle between $P Q$ and line $L$ (which passes through $P$ ) approaches $0,$ must $L$ be tangent to $C$ ?

Carson Merrill

Numerade Educator

Problem 32

Let $P(x)$ be a polynomial. If $a$ is a real number, then $P(x)$ can be expressed in the form $$P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\dots+a_{n}(x-a)^{n}$$ for some $n \geq 0 .$ If $\ell(x)=m(x-a)+b,$ show that the straight line $y=\ell(x)$ is tangent to the graph of $y=P(x)$ at $x=a$ provided $P(x)-\ell(x)=(x-a)^{2} Q(x),$ where $Q(x)$ is
a polynomial.

Carson Merrill

Numerade Educator