Question 1

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

Accepted Answer

A) Slope: $\frac { 1 } { 2 }$ ; y-intercept: 1 
B) Slope: $\frac { 3 } { 2 }$ ; y-intercept: $\frac { 1 } { 2 }$ 
C) Slope: $\frac { 3 } { 2 }$ ; y-intercept: 1 
D) Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3 
A) Slope: $\frac { 1 } { 2 }$ ; y-intercept: 1 
B) Slope: $\frac { 3 } { 2 }$ ; y-intercept: $\frac { 1 } { 2 }$ 
C) Slope: $\frac { 3 } { 2 }$ ; y-intercept: 1 
D) Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3

Question 2

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1,1).

Accepted Answer

A) -5 
B) 5 
C) 3 
D) 1 
A) -5 
B) 5 
C) 3 
D) 1

Question 3

Compute f (4,3)if $f ( x , y ) = 5 x y ^ { 3 }$ .

Accepted Answer

A) 108 
B) 192 
C) 960 
D) 540 
A) 108 
B) 192 
C) 960 
D) 540

Question 4

Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$ .

Accepted Answer

A) 1 
B)  $\frac { 1 } { 3 }$ 
C) Diverges 
D) 0 
A) 1 
B)  $\frac { 1 } { 3 }$ 
C) Diverges 
D) 0

Question 5

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 2 )$ of the given initial value problem.Round to one decimal place. $y ^ { t } = \frac { 2 x } { y } ; y ( 1 ) = 2$

Accepted Answer

A) 2.6 
B) 3.1 
C) 2.1 
D) 3.4 
A) 2.6 
B) 3.1 
C) 2.1 
D) 3.4

Question 6

$f ( x ) = \left\{ \begin{array} { l l } 
\frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \
0 & \text { if } x < 0
\end{array} \right.$ is a probability density function for a particular random variable X.Use integration to find $P ( X \geq 4 )$

Accepted Answer

A)  $e ^ { - 2 } - 1$ 
B)  $e ^ { 2 }$ 
C)  $e ^ { - 2 }$ 
D)  $1 + e ^ { 2 }$ 
A)  $e ^ { - 2 } - 1$ 
B)  $e ^ { 2 }$ 
C)  $e ^ { - 2 }$ 
D)  $1 + e ^ { 2 }$

Question 7

A wooded nature preserve supports populations of silver foxes and snowshoe hares.The growth rate for each population can be modeled by this pair of differential equations: $\begin{array} { l } 
\frac { d F } { d t } = - 0.24 F + 0.0012 F H \
\frac { d H } { d t } = 0.08 R - 0.005 F H
\end{array}$ where H is the number of hares and F is the number of foxes.Find equilibrium populations for this model.(Hint: These are the populations for which $\frac { d F } { d t } = \frac { d H } { d t } = 0$ .)

Accepted Answer

A) 12 foxes,170 hares 
B) 12 foxes,100 hares 
C) 30 foxes,220 hares 
D) 16 foxes,200 hares 
A) 12 foxes,170 hares 
B) 12 foxes,100 hares 
C) 30 foxes,220 hares 
D) 16 foxes,200 hares

Question 8

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 6 }$ the ratio in the atmosphere.Approximately how old is the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years.Round to the nearest whole year.

Accepted Answer

A) 17,190 years 
B) 14,812 years 
C) 34,380 years 
D) 14,842 years 
A) 17,190 years 
B) 14,812 years 
C) 34,380 years 
D) 14,842 years

Question 9

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }$

Accepted Answer

A) True 
 B)False

Question 10

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$ .

Accepted Answer

A)  $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$ 
B)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$ 
C)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
D)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
A)  $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$ 
B)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$ 
C)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$ 
D)  $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$

Question 11

A manufacturer will supply x units of a certain commodity to the market when the price is $p = S ( x )$ dollars per unit,where $S ( x ) = 21 + 2 x - 2.5 \sin \pi x$ . What is the average price per unit as the level of production increases from $x = 4$ to $x = 7 ?$ (Round to the nearest cent.)

Accepted Answer

A) $32.53 per unit 
B) $32.00 per unit 
C) $31.47 per unit 
D) $42.47 per unit 
A) $32.53 per unit 
B) $32.00 per unit 
C) $31.47 per unit 
D) $42.47 per unit

Question 12

Find the composite function $f ( g ( x ) )$ . $f ( u ) = \frac { 1 } { u }$ , $g ( x ) = x + 3$

Accepted Answer

A) f (g(x))= $1 + \frac { 3 } { x }$ 
B) f (g(x))= $\frac { 1 } { x + 3 }$ 
C) f (g(x))= $x + 3 + \frac { 1 } { x }$ 
D) f (g(x))= $\frac { 1 } { x } + 3$ 
A) f (g(x))= $1 + \frac { 3 } { x }$ 
B) f (g(x))= $\frac { 1 } { x + 3 }$ 
C) f (g(x))= $x + 3 + \frac { 1 } { x }$ 
D) f (g(x))= $\frac { 1 } { x } + 3$

Question 13

Simplify: $\frac { 3 ^ { 8 } } { 3 ^ { 7 } }$

Accepted Answer

A)  $\frac { 1 } { 9 }$ 
B) 3 
C)  $\frac { 1 } { 3 }$ 
D)  $3 ^ { 15 }$ 
A)  $\frac { 1 } { 9 }$ 
B) 3 
C)  $\frac { 1 } { 3 }$ 
D)  $3 ^ { 15 }$

Question 14

The second derivative test reveals that f (x)= x2 - 4 has

Accepted Answer

A) nothing significant at x = 4. 
B) a relative maximum at $x = \sqrt { 4 }$ . 
C) a point of inflection at $x = \sqrt { 4 }$ . 
D) a relative minimum at x = 0. 
A) nothing significant at x = 4. 
B) a relative maximum at $x = \sqrt { 4 }$ . 
C) a point of inflection at $x = \sqrt { 4 }$ . 
D) a relative minimum at x = 0.

Question 15

An object moving in a straight line has velocity $v ( t ) = t e ^ { \sqrt { t } }$ meters per second.Is it true that in the first 4 seconds the object will have travelled $4 e ^ { 2 } + 12$ meters?

Accepted Answer

A) True 
 B)False

Question 16

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 4$ .

Accepted Answer

A) Decreasing for $x - \frac { 5 } { 2 }$ B) Decreasing for all x C) Increasing for all x D) Decreasing for $x > - \frac { 5 } { 2 }$ and increasing for $x < - \frac { 5 } { 2 }$ A) Decreasing for $x - \frac { 5 } { 2 }$ B) Decreasing for all x C) Increasing for all x D) Decreasing for $x > - \frac { 5 } { 2 }$ and increasing for $x < - \frac { 5 } { 2 }$

Question 17

A ball is dropped from a height of H feet and bounces indefinitely,repeatedly rebounding to 76% of its previous height.If it travels a total distance of 80 feet,what is H? Round your answer to one decimal place.

Accepted Answer

A) 9.6 feet 
B) 30.4 feet 
C) 19.2 feet 
D) 10.9 feet 
A) 9.6 feet 
B) 30.4 feet 
C) 19.2 feet 
D) 10.9 feet

Question 18

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$ .

Accepted Answer

A)  $\frac { 1 } { 2 \ln 2 }$ 
B)  $- \frac { 1 } { 2 \ln 2 }$ 
C) Diverges 
D) 0 
A)  $\frac { 1 } { 2 \ln 2 }$ 
B)  $- \frac { 1 } { 2 \ln 2 }$ 
C) Diverges 
D) 0

Question 19

An investment will generate income continuously at the constant rate of $1,100 per year for 5 years.If the prevailing annual interest rate remains fixed at 6% compounded continuously,what is the present value of the investment?

Accepted Answer

A) $6,044.13 
B) $47.52 
C) $4,751.67 
D) $475.17 
A) $6,044.13 
B) $47.52 
C) $4,751.67 
D) $475.17

Question 20

$f ( x ) = \left\{ \begin{array} { l l } 
\frac { 1 } { 6 } & \text { if } 2 \leq x \leq 8 \
0 & \text { otherwise }
\end{array} \right.$ is a probability density function for a continuous random variable X. The expected value is 5.

Accepted Answer

A) True 
 B)False

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1,1).

Compute f (4,3)if $f ( x , y ) = 5 x y ^ { 3 }$ .

Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$ .

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 2 )$ of the given initial value problem.Round to one decimal place. $y ^ { t } = \frac { 2 x } { y } ; y ( 1 ) = 2$

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ is a probability density function for a particular random variable X.Use integration to find $P ( X \geq 4 )$

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 6 }$ the ratio in the atmosphere.Approximately how old is the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years.Round to the nearest whole year.

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }$

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$ .

A manufacturer will supply x units of a certain commodity to the market when the price is $p = S ( x )$ dollars per unit,where $S ( x ) = 21 + 2 x - 2.5 \sin \pi x$ . What is the average price per unit as the level of production increases from $x = 4$ to $x = 7 ?$ (Round to the nearest cent.)

Find the composite function $f ( g ( x ) )$ . $f ( u ) = \frac { 1 } { u }$ , $g ( x ) = x + 3$

Simplify: $\frac { 3 ^ { 8 } } { 3 ^ { 7 } }$

The second derivative test reveals that f (x)= x² - 4 has

An object moving in a straight line has velocity $v ( t ) = t e ^ { \sqrt { t } }$ meters per second.Is it true that in the first 4 seconds the object will have travelled $4 e ^ { 2 } + 12$ meters?

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 4$ .

A ball is dropped from a height of H feet and bounces indefinitely,repeatedly rebounding to 76% of its previous height.If it travels a total distance of 80 feet,what is H? Round your answer to one decimal place.

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$ .

An investment will generate income continuously at the constant rate of $1,100 per year for 5 years.If the prevailing annual interest rate remains fixed at 6% compounded continuously,what is the present value of the investment?

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 6 } & \text { if } 2 \leq x \leq 8 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a continuous random variable X. The expected value is 5.

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

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Exam 13: Mathematical Problem Set: Calculus, Algebra, and Optimization

Find the slope and y-intercept of the line whose equation is given. x2+y3=1\frac { x } { 2 } + \frac { y } { 3 } = 12x​+3y​=1

Find the slope of the tangent line to the curve x2+3xy−y2=3x ^ { 2 } + 3 x y - y ^ { 2 } = 3x2+3xy−y2=3 at the point (1,1).

Compute f (4,3)if f(x,y)=5xy3f ( x , y ) = 5 x y ^ { 3 }f(x,y)=5xy3 .

Evaluate ∫1∞dxx3\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }∫1∞​3x​dx​ .

Use Euler's method with the step size h=0.2h = 0.2h=0.2 to estimate the solution y(2)y ( 2 )y(2) of the given initial value problem.Round to one decimal place. yt=2xy;y(1)=2y ^ { t } = \frac { 2 x } { y } ; y ( 1 ) = 2yt=y2x​;y(1)=2

The given series converges. ∑k=1∞3k+5k5k+7k\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }∑k=1∞​5k+7k3k+5k​

Evaluate ∫x3(x2−3)7dx\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x∫x3(x2−3)7dx .

Find the composite function f(g(x))f ( g ( x ) )f(g(x)) . f(u)=1uf ( u ) = \frac { 1 } { u }f(u)=u1​ , g(x)=x+3g ( x ) = x + 3g(x)=x+3

Simplify: 3837\frac { 3 ^ { 8 } } { 3 ^ { 7 } }3738​

The second derivative test reveals that f (x)= x2 - 4 has

An object moving in a straight line has velocity v(t)=tetv ( t ) = t e ^ { \sqrt { t } }v(t)=tet​ meters per second.Is it true that in the first 4 seconds the object will have travelled 4e2+124 e ^ { 2 } + 124e2+12 meters?

Find the intervals of increase and decrease for the function f(x)=x2+5x−4f ( x ) = x ^ { 2 } + 5 x - 4f(x)=x2+5x−4 .

A ball is dropped from a height of H feet and bounces indefinitely,repeatedly rebounding to 76% of its previous height.If it travels a total distance of 80 feet,what is H? Round your answer to one decimal place.

Evaluate ∫2∞1xln⁡xdx\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x∫2∞​xlnx​1​dx .

An investment will generate income continuously at the constant rate of $1,100 per year for 5 years.If the prevailing annual interest rate remains fixed at 6% compounded continuously,what is the present value of the investment?

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Filters

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1,1).

Compute f (4,3)if $f ( x , y ) = 5 x y ^ { 3 }$ .

Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$ .

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 2 )$ of the given initial value problem.Round to one decimal place. $y ^ { t } = \frac { 2 x } { y } ; y ( 1 ) = 2$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }$

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$ .

Find the composite function $f ( g ( x ) )$ . $f ( u ) = \frac { 1 } { u }$ , $g ( x ) = x + 3$

Simplify: $\frac { 3 ^ { 8 } } { 3 ^ { 7 } }$

The second derivative test reveals that f (x)= x² - 4 has

An object moving in a straight line has velocity $v ( t ) = t e ^ { \sqrt { t } }$ meters per second.Is it true that in the first 4 seconds the object will have travelled $4 e ^ { 2 } + 12$ meters?

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 4$ .

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$ .