Exam 6: Additional Topics in Integration

Suppose the length of time that it takes a person to complete a hedgerow maze is measured by a random variable X that is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 4 } x e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is the number of minutes a randomly selected person takes to complete the maze. Find the probability that a randomly chosen person will take less than 4 minutes to complete the maze.

(Short Answer)

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Question 101

From time t = 0 to t = 5 an object's speed is given by the function $s ( t ) = t e ^ { 3 t }$ . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

(Multiple Choice)

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Question 102

A 2-hour movie runs continuously at a local theater. You leave for the theater without first checking the show times. Use an appropriate uniform density function to find the probability that you will arrive at the theater within 4 minutes of the start of the film (before or after). Round to the nearest hundredth.

(Multiple Choice)

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Question 103

If, after t hours on the job, a factory worker can produce $10 t e ^ { - 0.02 t }$ units per hour, then the worker produces 130 units during the first 5 hours.

(True/False)

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Question 104

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

(Multiple Choice)

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Question 105

$f ( x ) = \left\{ \begin{array} { l l } 4 e ^ { - 4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

(True/False)

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Question 106

Evaluate $\int x ^ { 3 } e ^ { x ^ { 2 } } d x$ .

(Short Answer)

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Question 107

Evaluate $\int x e ^ { x / 5 } d x$

(Multiple Choice)

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Question 108

Suppose the length of time, x, that it takes a chimpanzee to solve a simple puzzle is measured by a random variable X that is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } \frac { 2 } { 3 } e ^ { - 2 x / 3 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is in minutes. Find the probability that a randomly chosen chimpanzee will take more than 12 minutes to solve the puzzle.

(Multiple Choice)

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Question 109

After t seconds, an object is moving at the speed of $t e ^ { 1 - t }$ meters per second. If the object begins at 0 when t = 0, then the distance the object travels as a function of time is expressed $s ( t ) = - e ^ { 1 - t } ( t + 1 )$ .

(True/False)

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Question 110

Evaluate $\int x ^ { 2 } ( \ln x ) ^ { 2 } d x$ .

(Short Answer)

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Question 111

From time t = 0 to t = 5 an object's speed is given by the function $s ( t ) = t e ^ { 3 t }$ . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

If, after t hours on the job, a factory worker can produce $10 t e ^ { - 0.02 t }$ units per hour, then the worker produces 130 units during the first 5 hours.

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

$f ( x ) = \left\{ \begin{array} { l l } 4 e ^ { - 4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

Evaluate $\int x ^ { 3 } e ^ { x ^ { 2 } } d x$ .

Evaluate $\int x e ^ { x / 5 } d x$

After t seconds, an object is moving at the speed of $t e ^ { 1 - t }$ meters per second. If the object begins at 0 when t = 0, then the distance the object travels as a function of time is expressed $s ( t ) = - e ^ { 1 - t } ( t + 1 )$ .

Evaluate $\int x ^ { 2 } ( \ln x ) ^ { 2 } d x$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Calculus of Several Variables

Appendix: Algebra Review

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Exam 6: Additional Topics in Integration

From time t = 0 to t = 5 an object's speed is given by the function s(t)=te3ts ( t ) = t e ^ { 3 t }s(t)=te3t . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

If, after t hours on the job, a factory worker can produce 10te−0.02t10 t e ^ { - 0.02 t }10te−0.02t units per hour, then the worker produces 130 units during the first 5 hours.

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

f(x)={4e−4x for x≥00 for x<0f ( x ) = \left\{ \begin{array} { l l } 4 e ^ { - 4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.f(x)={4e−4x0​ for x≥0 for x<0​ is a probability density function.

Evaluate ∫x3ex2dx\int x ^ { 3 } e ^ { x ^ { 2 } } d x∫x3ex2dx .

Evaluate ∫xex/5dx\int x e ^ { x / 5 } d x∫xex/5dx

After t seconds, an object is moving at the speed of te1−tt e ^ { 1 - t }te1−t meters per second. If the object begins at 0 when t = 0, then the distance the object travels as a function of time is expressed s(t)=−e1−t(t+1)s ( t ) = - e ^ { 1 - t } ( t + 1 )s(t)=−e1−t(t+1) .

Evaluate ∫x2(ln⁡x)2dx\int x ^ { 2 } ( \ln x ) ^ { 2 } d x∫x2(lnx)2dx .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

From time t = 0 to t = 5 an object's speed is given by the function $s ( t ) = t e ^ { 3 t }$ . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

If, after t hours on the job, a factory worker can produce $10 t e ^ { - 0.02 t }$ units per hour, then the worker produces 130 units during the first 5 hours.

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

$f ( x ) = \left\{ \begin{array} { l l } 4 e ^ { - 4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

Evaluate $\int x ^ { 3 } e ^ { x ^ { 2 } } d x$ .

Evaluate $\int x e ^ { x / 5 } d x$

After t seconds, an object is moving at the speed of $t e ^ { 1 - t }$ meters per second. If the object begins at 0 when t = 0, then the distance the object travels as a function of time is expressed $s ( t ) = - e ^ { 1 - t } ( t + 1 )$ .

Evaluate $\int x ^ { 2 } ( \ln x ) ^ { 2 } d x$ .