Question 1

Find the limit. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 10 i } { n } \right) \left( \frac { 8 } { n } \right)$

Accepted Answer

A)  20 
B)  18 
C)  36 
D)  80 
E)  40 
A)  20 
B)  18 
C)  36 
D)  80 
E)  40

Question 2

$\text { Find the area of the shaded region for the function } y = \frac { 2 } { \sqrt { 4 - x ^ { 2 } } } \text {. }$

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 5 \pi } { 6 }$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 5 \pi } { 6 }$

Question 3

$\text { The rate of depreciation } d V / d t \text { of a machine is inversely proportional to the square }$ of $t + 2$, where $V$ is the value of the machine $t$ years after it was purchased. The initial value of the machine was $\$ 500,000$, and its value decreased $\$ 100,000$ in the first year. Estimate its value after 6 years. Round your answer to the nearest integer.

Accepted Answer

A)  $\$ 275,000$ 
B)  $\$ 475,000$ 
C)  $\$ 550,000$ 
D)  $\$ 350,000$ 
E)  $\$ 500,000$ 
A)  $\$ 275,000$ 
B)  $\$ 475,000$ 
C)  $\$ 550,000$ 
D)  $\$ 350,000$ 
E)  $\$ 500,000$

Question 4

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 1 } 2 \cos ( \pi x ) d x$ is less than $0.00001$ using Simpson's Rule.

Accepted Answer

A)  48 
B)  8 
C)  11 
D)  14 
E)  19 
A)  48 
B)  8 
C)  11 
D)  14 
E)  19

Question 5

Find the indefinite integral. $\int \frac { x ^ { 3 } } { 2 x ^ { 4 } + 1 } d x$

Accepted Answer

A)  $\frac { 1 } { 8 } \ln \left| 2 x ^ { 4 } + 1 \right| + C$ 
B)  $\frac { 1 } { 2 x ^ { 4 } + 1 } + C$ 
C)  $\ln \left| 2 x ^ { 4 } - 1 \right| + C$ 
D)  $\ln \left| 2 x ^ { 4 } + 1 \right| + C$ 
E)  $\frac { 1 } { 16 } \ln \left| 2 x ^ { 4 } + 1 \right| + C$ 
A)  $\frac { 1 } { 8 } \ln \left| 2 x ^ { 4 } + 1 \right| + C$ 
B)  $\frac { 1 } { 2 x ^ { 4 } + 1 } + C$ 
C)  $\ln \left| 2 x ^ { 4 } - 1 \right| + C$ 
D)  $\ln \left| 2 x ^ { 4 } + 1 \right| + C$ 
E)  $\frac { 1 } { 16 } \ln \left| 2 x ^ { 4 } + 1 \right| + C$

Question 6

The diagram below shows upper and lower sums for the function $f ( x ) = 2 \sqrt { 1 - x ^ { 3 / 2 } }$ using 4 subintervals. Use upper and lower sums to approximate the area of the region using the 4 subintervals.

Accepted Answer

A)  lower: $1.166$; upper: $1.666$ 
B)  lower: $4.664$; upper: $6.664$ 
C)  lower: $2.332$; upper: $3.332$ 
D)  lower: $0.916$; upper: $1.916$ 
A)  lower: $1.166$; upper: $1.666$ 
B)  lower: $4.664$; upper: $6.664$ 
C)  lower: $2.332$; upper: $3.332$ 
D)  lower: $0.916$; upper: $1.916$

Question 7

Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function and the x-axis over the interval $\left[ 0 , \frac { \pi } { 6 } \right] \text {. Round your }$ answer to four decimal places.

Accepted Answer

A) 0.0419 
B) 0.1319 
C) 0.1219 
D) 0.0819 
E) 0.0119 
A) 0.0419 
B) 0.1319 
C) 0.1219 
D) 0.0819 
E) 0.0119

Find the limit. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 10 i } { n } \right) \left( \frac { 8 } { n } \right)$

$\text { Find the area of the shaded region for the function } y = \frac { 2 } { \sqrt { 4 - x ^ { 2 } } } \text {. }$ $\text { Find the area of the shaded region for the function } y = \frac { 2 } { \sqrt { 4 - x ^ { 2 } } } \text {. }$

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 1 } 2 \cos ( \pi x ) d x$ is less than $0.00001$ using Simpson's Rule.

Find the indefinite integral. $\int \frac { x ^ { 3 } } { 2 x ^ { 4 } + 1 } d x$

Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function and the x-axis over the interval $\left[ 0 , \frac { \pi } { 6 } \right] \text {. Round your }$ answer to four decimal places.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 5: Antiderivatives and Indefinite Integration

Find the limit. lim⁡n→∞∑i=1n(10in)(8n)\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 10 i } { n } \right) \left( \frac { 8 } { n } \right)limn→∞​∑i=1n​(n10i​)(n8​)

Find the area of the shaded region for the function y=24−x2. \text { Find the area of the shaded region for the function } y = \frac { 2 } { \sqrt { 4 - x ^ { 2 } } } \text {. } Find the area of the shaded region for the function y=4−x2​2​.

Find the smallest n such that the error estimate in the approximation of the definite integral ∫012cos⁡(πx)dx\int _ { 0 } ^ { 1 } 2 \cos ( \pi x ) d x∫01​2cos(πx)dx is less than 0.000010.000010.00001 using Simpson's Rule.

Find the indefinite integral. ∫x32x4+1dx\int \frac { x ^ { 3 } } { 2 x ^ { 4 } + 1 } d x∫2x4+1x3​dx

The diagram below shows upper and lower sums for the function f(x)=21−x3/2f ( x ) = 2 \sqrt { 1 - x ^ { 3 / 2 } }f(x)=21−x3/2​ using 4 subintervals. Use upper and lower sums to approximate the area of the region using the 4 subintervals.

Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function and the x-axis over the interval [0,π6]. Round your \left[ 0 , \frac { \pi } { 6 } \right] \text {. Round your }[0,6π​]. Round your answer to four decimal places.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the limit. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 10 i } { n } \right) \left( \frac { 8 } { n } \right)$

$\text { Find the area of the shaded region for the function } y = \frac { 2 } { \sqrt { 4 - x ^ { 2 } } } \text {. }$ $\text { Find the area of the shaded region for the function } y = \frac { 2 } { \sqrt { 4 - x ^ { 2 } } } \text {. }$

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 1 } 2 \cos ( \pi x ) d x$ is less than $0.00001$ using Simpson's Rule.

Find the indefinite integral. $\int \frac { x ^ { 3 } } { 2 x ^ { 4 } + 1 } d x$

Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function and the x-axis over the interval $\left[ 0 , \frac { \pi } { 6 } \right] \text {. Round your }$ answer to four decimal places.