## Questions and Answers for Single Variable Calculus

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 55

Suppose that a ship is moving at a speed of 30km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) We need to express the distance $s$...

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 56

Find the area bounded by the curves $y = \arctan 3x$ and $\displaystyle y = \frac{1}{2} x$ by approximating the $x$-coordiantes of the points of intersections. Based from the...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 10

The given graph shows the amount $f(t)$ of the drug in the blood stream of a patient after $t$ hours. Suppose a patient receives a $150 \, mg$ injection of a drug every 4 hours. Find...

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 18

Evaluate $\displaystyle \int e^{-\theta} \cos 2 \theta d \theta$...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 20

Show that the statement $\lim \limits_{x \to 0} \sqrt[3]{x} = 0$...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 12

The graph of the position function of two runners $A$ and $B$ is shown below. Suppose that the runners finish a 100$m$ race in a tie. a.) Describe and compare how the runners run the race....

Single Variable Calculus

### Single Variable Calculus, Chapter 5, Review Exercises, Section Review Exercises, Problem 34

Determine the derivative of the function $\displaystyle F(x) = \int^1_x \sqrt{t + \sin t} dt$...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 18

Evaluate the function $\displaystyle \lim \limits_{x \to -1} \frac{x^2 - 2x}{x^2 - x - 2}$ at the given numbers $x = 0, -0.5, -0.9, -0.95, -0.99, -2, -1.5, -1.1, -1.01, -1.001$ and guess the...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 56

Assume that an airplane is flying at a speed of $350mi/h$ at an altitude of one mile and passes directly over a radar station at time $t = 0$. (a) We need to express the horizontal distance...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 44

Suppose that a cylindrical tank holds a 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 20

The temperature readings T (in $^\circ$...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.2, Section 1.2, Problem 12

Based from the past experience of the manager of the weekend flea market, that if he charges $x$ dollars for a rental space at the market, then the number $y$ of spaces he can rent is given by...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 41

Determine the value of the constant $c$ that makes the function $...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 57

The Heaviside function $H$ is defined by $H(t)$ = $\left\{\begin{array}{cccc} 0 & if & t < 0 \\ 1 & if & t\geq 0 \end{array} \right.$ Suppose that it is used in the...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 48

Use the Intermediate Value Theorem to show that $\sqrt[3]{x} = 1 - x$ has root on the interval $(0, 1)$ Let $f(x) = \sqrt[3]{x}+x-1$ Based from the definition of Intermediate Value Theorem,...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 51

(a) Show that the equation $\cos x = x^3$ has at least one real root. (b) Determine an inteval of length 0.01 that contains a root using a calculator. (a) Let $f(x) = x^3 - \cos x$ Based from...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 63

(a) If $f$ and $g$...

Single Variable Calculus

### Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 8

Determine the general indefinite integral $\displaystyle \int \left( y^3 + 1.8y^2 - 2.4y \right) dy$...

Single Variable Calculus

### Single Variable Calculus, Chapter 7, 7.3-1, Section 7.3-1, Problem 66

Determine the equation of the tangent line to the curve $y = e^{-x}$ that is perpendicular to the line $2x - y = 8$ Since the tangent line is perpendicular to the tangent line, its slope is...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 32

If $\displaystyle f \left( \frac{\pi}{3} \right) = 4$ and $\displaystyle f'\left( \frac{\pi}{3} \right) = -2$, and let $g(x) = f(x) \sin x$ and $\displaystyle h(x) = \frac{\cos x}{f(x)} $....

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 34

Determine the integral $\displaystyle \int \tan^2 x \sec x dx$...

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 42

Determine the integral $\displaystyle \int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx$...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 47

Find the limit $\displaystyle \lim\limits_{x \to \pi/4} \left( \frac{1 - \tan x}{\sin x - \cos x}\right)$...

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 48

Determine the integral $\displaystyle \int \frac{dx}{\cos x - 1}x$...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 21

Use the precise definition of a limit to prove the statement $\lim \limits_{x \to 2} (x^2 - 3x) = -2$...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 22

Show that the statement $\displaystyle \lim \limits_{x \to 4^+} \frac{2}{\sqrt{x - 4}} = \infty$ using the precise definition of a limit. Based from the definition, we let $M > 0$. So,...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 24

Suppose that $\qquad f(x) = \left\{ \begin{array}{ccc} 2x - x^2 & \text{ if } & 0 \leq x \leq 2 \\ 2 - x & \text{ if } & 2 < x \leq 3 \\ x - 4 & \text{ if } & 3 < x < 4 \\ \pi & \text{ if } & x \geq 4 \end{array} \right. $...

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 60

Use a graph of $\displaystyle \int^2_0 \sin 2 \pi x \cos 5 \pi x dx$ to guess the value of the integral. Then prove that your guess is correct by using Calculus. Based from the...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 26

Prove that $\displaystyle g(x) = \frac{\sqrt{x^2 - 9}}{x^2 - 2}$...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 27

Use the Intermediate Value Theorem to show that $2x^3 + x^2 + 2 = 0$ has a root in the interval $(-2, -1)$ Let $f(x) = 2x^3 + x^2 + 2$ Based from the definition of Intermediate Value Theorem,...

Single Variable Calculus

### Single Variable Calculus, Chapter 8, 8.4, Section 8.4, Problem 52

Find the indefinite integral $\displaystyle \int \sec^4 \left( \frac{x}{2} \right) dx$. Illustrate by graphing both the integrand and its antiderivative (taking $c = 0$). Let...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 8

Determine the equation of the tangent line to the curve $\displaystyle y = \frac{2x}{(x + 1)^2} $...

Single Variable Calculus

### Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 46

Find the derivative of the function $y = x^{\cos x}$...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 17

State whether the function is even, odd or neither even nor odd. $ \begin{equation} \begin{aligned} \text{ a)} f(x) =& 2x^5 - 3x^2 + 2\\ f(-x) =& 2 (-x)^5 - 3 (-x)^2 + 2\\ f(-x) =& -2x^5 - 3x^2 + 2\\ \end{aligned} \end{equation} $...

Single Variable Calculus

### Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 48

Find the derivative of the function $y = \sqrt{x^x}$...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 31

$\displaystyle \lim_{h \to 0} \frac{(1 + h)^{10} - 1}{h}$ represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in this case. Using the limit equation for...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 41

Determine the derivative of the function $y = \sqrt{x+\sqrt{x}}$...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 36

$\displaystyle \lim_{t \to 1} \frac{t^4 + t - 2}{t - 1}$ represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in this case. Using the slope equation,...

Single Variable Calculus

### Single Variable Calculus, Chapter 4, 4.1, Section 4.1, Problem 72

The equation $f(x) = ax^3 + bx^2 + cx + d$, where $a \neq 0$ represents a cubic function (Polynomial of degree 3). a.) Prove that a cubic function can have one, two or more critical number(s)....

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 16

Draw the graph of a function $f$ that satifies all of the given conditions. The graph of $...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 52

Determine whether $f'(0)$...

Single Variable Calculus

### Single Variable Calculus, Chapter 6, 6.1, Section 6.1, Problem 20

Sketch the region enclosed by the curves $4x + y^2 = 12$...

Single Variable Calculus

### Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 62

Suppose that $h(x) = \sqrt{4 + 3 f(x)}$...

Single Variable Calculus

### Single Variable Calculus, Chapter 6, 6.1, Section 6.1, Problem 32

Evaluate $\displaystyle \int^4_0 |\sqrt{x + 2} - x| dx$...

Single Variable Calculus

### Single Variable Calculus, Chapter 6, 6.1, Section 6.1, Problem 44

The equation $b(t) = 2200 + 52.3 t + 0.74t^2$ represents the birth rate of a population of people per year and the death rate $d(t) = 1460 + 28.8t$ people per year. Find the area between the...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 3

Determine the $\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1)$...

Single Variable Calculus

### Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 6

Determine the Given: $\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5$...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 18

Express the horizontal distance x traveled by the airplane and its altitude y as a function of time in minutes when an airplane takes off from an airport and lands an hour later at another...

Single Variable Calculus

### Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 23

Evaluate the difference quotient $\displaystyle \frac{f (3+h)-f(3)}{h}$...

Single Variable Calculus

### Single Variable Calculus, Chapter 7, 7.7, Section 7.7, Problem 10

Show that $\cos h x - \sin h x = e^{-x}$...

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