Question 1

Evaluate the sum using the summation formula and property. 
 $\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$  
A)3,015
B)3,315
C)3,215
D)2,915
E)3,115

Accepted Answer

E

Question 2

Evaluate the sum using the summation formula and property. 
 $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 2 } + j \right)$  
A)2,980
B)3,080
C)2,880
D)3,180
E)3,280

Accepted Answer

The sum can be split into two separate sums: $\sum_{j=1}^{20} j^2 + \sum_{j=1}^{20} j$. Using the formulas $\sum_{j=1}^{n} j^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum_{j=1}^{n} j = \frac{n(n+1)}{2}$, we calculate each part: $\sum_{j=1}^{20} j^2 = \frac{20 \times 21 \times 41}{6} = 2870$ and $\sum_{j=1}^{20} j = \frac{20 \times 21}{2} = 210$. Adding these gives $2870 + 210 = 3080$.

Question 3

Find  $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right)\right]\left(\frac{1}{n}\right)$
A) $\lim_{n\rightarrow \infty } S ( n ) = 3$
B) $\lim _{n\rightarrow \infty }S ( n ) = 7$
C) $\lim_{n\rightarrow \infty } S ( n ) = 4$
D)  $\lim_{n\rightarrow\infty } S ( n ) = 5$
E)  $\lim_{n\rightarrow \infty } S ( n ) = 6$

Accepted Answer

The expression is a Riemann sum for the integral $\int_0^1 (4 - 2x) \, dx$, which evaluates to 3.

Question 4

Evaluate the sum using the summation formula and property.   $\sum _ { i = 1 } ^ { 45 } 4$  
A)160
B)200
C)190
D)180
E)170

Accepted Answer

The sum of a constant $k$ over $n$ terms in a summation is given by $n \times k$. Here, $k = 4$ and $n = 45$, so the sum is $45 \times 4 = 180$.

Question 5

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 3 x - 4&[2,5]\
\end{array}$ 
A)A = 21.5 square units
B)A = 21 square units
C)A = 19.5 square units
D)A = 20 square units
E)A = 20.5 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 6

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = - 2 x + 5&[0,2]\
\end{array}$ 
A)A = 7 square units
B)A = 8 square units
C)A = 5square units
D)A = 4 square units
E)A = 6 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 7

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
  $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f(x)=3 x+1&[0,2]
\end{array}$ 
A)A = 7 square units
B)A = 11 square units
C)A = 10 square units
D)A = 8 square units
E)A = 9 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 8

Evaluate the sum using the summation formula and property. 
$\sum _ { i = 1 } ^ { 10 } i ^ { 3 }$ 
A)3,125
B)2,825
C)3,225
D)2,925
E)3,025

Accepted Answer

The answer of Evaluate the sum using the summation formula...

Question 9

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 9 - x ^ { 3 }&[1,2]\
\end{array}$
 
A)A = 5.375 square units
B)A = 5.5 square units
C)A = 5.25 square units
D)A = 5.625 square units
E)A = 5.125 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 10

Evaluate the sum using the summation formula and property. 
 $\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$  
A)9,355
B)9,455
C)9,655
D)9,255
E)9,555

Accepted Answer

The sum of squares formula is $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$. For $n = 30$, it becomes $\frac{30 \times 31 \times 61}{6} = 9,455$.

Question 11

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 66 - x ^ { 3 }&[1,4]\
\end{array}$ 
A)A = 85.5 square units
B)A = 36.75 square units
C)A = 231.75 square units
D)A = 183 square units
E)A = 134.25 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 12

Evaluate the sum using the summation formula and property. 
 $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$  
A)89,150
B)88,950
C)89,250
D)88,850
E)89,050

Accepted Answer

The given sum can be split into two separate sums using the summation property: $\sum _ { j = 1 } ^ { 25 } j ^ { 3 } - \sum _ { j = 1 } ^ { 25 } 3 j ^ { 2 }$. The formula for the sum of cubes is $\frac{n^2(n+1)^2}{4}$ and for the sum of squares is $\frac{n(n+1)(2n+1)}{6}$. Substituting $n=25$, we get $\frac{25^2(26)^2}{4} - 3\left(\frac{25(26)(51)}{6}\right)$, which simplifies to 89,050.

Question 13

Rewrite the sum as a rational function S(n).  $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$  
A) $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 2 } }$
B) $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 4 } }$
C) $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 5 } }$
D) $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n ^ { 3 } }$
E) $S ( n ) = \frac { n ^ { 2 } + 2 n + 1 } { 4 n }$

Accepted Answer

The answer of Rewrite the sum as a rational function...

Question 14

Evaluate the sum using the summation formula and property.   $\sum _ { i = 1 } ^ { 60 } 7$  
A)410
B)400
C)420
D)430
E)440

Accepted Answer

The sum of a constant $k$ over $n$ terms in a summation is given by $n \times k$. Here, $k = 7$ and $n = 60$, so the sum is $60 \times 7 = 420$.

Question 15

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 8 x - x ^ { 3 } &[0,1]\
\end{array}$

A)A = 4.5 square units
B)A = 3 square units
C)A = 4 square units
D)A = 3.5 square units
E)A = 3.75 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 16

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 6 - x ^ { 2 }&[2,5]\
\end{array}$ 
A)A = 12.33 square units
B)A = 10.33 square units
C)A = 9.33 square units
D)A = 13.33 square units
E)A = 21 square units

Accepted Answer

The answer of Use the limit process to find the...

Question 17

Find $\lim  _ { n \rightarrow \infty }S ( n )$ . 
 $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 4 } }$  
A) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 6 }$
B) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 5 }$
C) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 4 }$
D) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 8 }$
E) $\lim _ { n \rightarrow \infty } S ( n ) = \frac { 1 } { 7 }$

Accepted Answer

The answer of Find \(\lim  _ { n \rightarrow...

Question 18

Rewrite the sum as a rational function S(n). $\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }$ 
A)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 5 } }$
B)  $S ( n ) = \frac { n + 1 } { 2 n ^ { 6 } }$
C) $S ( n ) = \frac { n + 1 } { 2 n ^ { 3 } }$ 
D) $S ( n ) = \frac { n + 1 } { 2 n ^ { 7 } }$
E) $S ( n ) = \frac { n + 1 } { 2 n ^ { 4 } }$

Accepted Answer

The answer of Rewrite the sum as a rational function...

Question 19

Evaluate the sum using the summation formula and property. 
 $\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$  
A)1,465
B)1,565
C)1,165
D)1,365
E)1,265

Accepted Answer

The answer of Evaluate the sum using the summation formula...

Question 20

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = x ^ { 2 } + 2&[0,1]\
\end{array}$ 
A)A = 2 square units
B)A = 3.33 square units
C)A = 3 square units
D)A = 2.33 square units
E)A = 2.67 square units

Accepted Answer

The area under the curve $ f(x) = x^2 + 2 $ from $ x = 0 $ to $ x = 1 $ is calculated using the definite integral: $\int_{0}^{1} (x^2 + 2) \, dx = \left[\frac{x^3}{3} + 2x\right]_{0}^{1} = \frac{1}{3} + 2 = \frac{7}{3} = 2.33$ square units.

Quiz 12: Limits and An Introduction To Calculus

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 2 } + j \right)$

Find $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right) \right]\left(\frac{1}{n}\right)$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 45 } 4$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 3 x - 4&[2,5]\\\end{array}$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = - 2 x + 5&[0,2]\\\end{array}$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f(x) =3 x+1&[0,2]\end{array}$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 10 } i ^ { 3 }$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 9 - x ^ { 3 }&[1,2]\\\end{array}$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 66 - x ^ { 3 }&[1,4]\\\end{array}$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$

Rewrite the sum as a rational function S(n) . $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 60 } 7$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 8 x - x ^ { 3 } &[0,1]\\\end{array}$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = 6 - x ^ { 2 }&[2,5]\\\end{array}$

Find $\lim _ { n \rightarrow \infty }S ( n )$ . $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 4 } }$

Rewrite the sum as a rational function S(n) . $\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }$

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f ( x ) = x ^ { 2 } + 2&[0,1]\\\end{array}$

Quiz 12: Limits and An Introduction To Calculus

Evaluate the sum using the summation formula and property. ∑k=110(k3+9) \sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right) ∑k=110​(k3+9)

Evaluate the sum using the summation formula and property. ∑j=120(j2+j) \sum _ { j = 1 } ^ { 20 } \left( j ^ { 2 } + j \right) ∑j=120​(j2+j)

Find lim⁡n→∞S(n) ∑i=1n[4−2(in) ](1n) \lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right) \right]\left(\frac{1}{n}\right) limn→∞​S(n) ∑i=1n​[4−2(ni​) ](n1​)

Evaluate the sum using the summation formula and property. ∑i=1454\sum _ { i = 1 } ^ { 45 } 4∑i=145​4

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x) =3x+1[0,2]\begin{array} {l } \text {Function }& \text {Interval }\\f(x) =3 x+1&[0,2]\end{array}Function f(x) =3x+1​Interval [0,2]​

Evaluate the sum using the summation formula and property. ∑i=110i3\sum _ { i = 1 } ^ { 10 } i ^ { 3 }∑i=110​i3

Evaluate the sum using the summation formula and property. ∑i=130i2\sum _ { i = 1 } ^ { 30 } i ^ { 2 }∑i=130​i2

Evaluate the sum using the summation formula and property. ∑j=125(j3−3j2) \sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right) ∑j=125​(j3−3j2)

Rewrite the sum as a rational function S(n) . ∑i=1ni3n3\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }∑i=1n​n3i3​

Evaluate the sum using the summation formula and property. ∑i=1607\sum _ { i = 1 } ^ { 60 } 7∑i=160​7

Find lim⁡n→∞S(n) \lim _ { n \rightarrow \infty }S ( n ) limn→∞​S(n) . ∑i=1ni3n4\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 4 } }∑i=1n​n4i3​

Rewrite the sum as a rational function S(n) . ∑i=1nin4\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }∑i=1n​n4i​

Evaluate the sum using the summation formula and property. ∑k=135(2k+3) \sum _ { k = 1 } ^ { 35 } ( 2 k + 3 ) ∑k=135​(2k+3)

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 10 } \left( k ^ { 3 } + 9 \right)$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 20 } \left( j ^ { 2 } + j \right)$

Find $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right) \right]\left(\frac{1}{n}\right)$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 45 } 4$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $\begin{array} {l } \text {Function }& \text {Interval }\\f(x) =3 x+1&[0,2]\end{array}$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 10 } i ^ { 3 }$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 30 } i ^ { 2 }$

Evaluate the sum using the summation formula and property. $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$

Rewrite the sum as a rational function S(n) . $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 3 } }$

Evaluate the sum using the summation formula and property. $\sum _ { i = 1 } ^ { 60 } 7$

Find $\lim _ { n \rightarrow \infty }S ( n )$ . $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 4 } }$

Rewrite the sum as a rational function S(n) . $\sum _ { i = 1 } ^ { n } \frac { i } { n ^ { 4 } }$

Evaluate the sum using the summation formula and property. $\sum _ { k = 1 } ^ { 35 } ( 2 k + 3 )$