Question 1

Solve the problem.
-The second derivative of a person's body temperature, with respect to the dosage of  xxx  milligrams of a drug, is given by  D′(x) =8x+5D^{\prime}(x) =\frac{8}{x+5}D′(x) =x+58​ . One milligram raises the temperature  1.5∘C1.5^{\circ} \mathrm{C}1.5∘C . Find the function  D(x) D(x) D(x)   giving the total change in temperature as a function of  xxx .

Accepted Answer

A)    D(x) =ln⁡∣8x+5∣+12.8\mathrm{D}(\mathrm{x}) =\ln \left|\frac{8}{\mathrm{x}+5}\right|+12.8D(x) =lnc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​x+58​c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​+12.8 
B)    D(x) =8ln⁡∣x+5∣−12.8\mathrm{D}(\mathrm{x}) =8 \ln |\mathrm{x}+5|-12.8D(x) =8ln∣x+5∣−12.8 
C)    D(x) =8ln⁡∣x+5∣+1.5D(x) =8 \ln |x+5|+1.5D(x) =8ln∣x+5∣+1.5 
D)    D(x) =ln⁡∣8x+5∣−1.5\mathrm{D}(\mathrm{x}) =\ln \left|\frac{8}{\mathrm{x}+5}\right|-1.5D(x) =lnc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​x+58​c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​−1.5

Question 2

Find the integral.
- ∫2e4zdz\int 2 e^{4 z} d z∫2e4zdz

Accepted Answer

A)    12e4z+C\frac{1}{2} e^{4 z}+C21​e4z+C 
B)    14e4z+C\frac{1}{4} e^{4 z}+C41​e4z+C 
C)    4e4z+C4 e^{4 z}+C4e4z+C 
D)    2e4z+C2 e^{4 z}+C2e4z+C

Question 3

Solve the problem.
-Suppose the supply function of a certain item is given by  S(x) =2x+7S(x) =2 x+7S(x) =2x+7  and the demand function is  D(x) =27−x2/3D(x) =27-x^{2 / 3}D(x) =27−x2/3 . Find the producer's surplus.

Accepted Answer

A)    325\frac{32}{5}532​ 
B)   32
C)   64
D)    645\frac{64}{5}564​

Question 4

Find the particular solution of the differential equation.
- dydx−xy−x=0;y=10\frac{d y}{d x}-x y-x=0 ; y=10dxdy​−xy−x=0;y=10  when  x=1x=1x=1

Accepted Answer

A)    y=−12+(112) ex2−1y=-\frac{1}{2}+\left(\frac{11}{2}\right)  e^{x^{2}-1}y=−21​+(211​) ex2−1 
B)    y=−12+11e−x2\mathrm{y}=-\frac{1}{2}+11 \mathrm{e}^{-\mathrm{x}^{2}}y=−21​+11e−x2 
C)    y=−1+11e−x2y=-1+11 e^{-x^{2}}y=−1+11e−x2 
D)    y=−1+11e(x2−1) /2y=-1+11 e^{\left(x^{2}-1\right)  / 2}y=−1+11e(x2−1) /2

Question 5

Find the integral.
- ∫8(y−9) 3dy\int \frac{8}{(y-9) ^{3}} d y∫(y−9) 38​dy

Accepted Answer

A)    2(y−9) 4+C\frac{2}{(y-9) ^{4}}+C(y−9) 42​+C 
B)    −2(y−9) 4+C\frac{-2}{(y-9) ^{4}}+C(y−9) 4−2​+C 
C)    4(y−9) 2+C\frac{4}{(y-9) ^{2}}+C(y−9) 24​+C 
D)    −4(y−9) 2+C\frac{-4}{(y-9) ^{2}}+C(y−9) 2−4​+C

Question 6

Solve the problem.
-The rate of infection of a disease (in people per month)  is given by the function  f′(t) =900tt2+1f^{\prime}(t) =\frac{900 t}{t^{2}+1}f′(t) =t2+1900t​ , where  ttt  is the time in months since the disease broke out. Find the total number of infected people over the first 5 months of the disease.

Accepted Answer

A)   806
B)   1466
C)   2932
D)   1613

Question 7

Use the definite integral to find the area between the  xxx -axis and the graph of  f(x) f(x) f(x)   over the indicated interval.
- f(x) =2x+7;[1,5]f(x) =2 x+7 ;[1,5]f(x) =2x+7;[1,5]

Accepted Answer

A)   52
B)   9
C)   26
D)   18

Question 8

Provide an appropriate response.
-If  r(t) r(t) r(t)   is the rate of change of revenue, then  ∫abr(t) dt\int_{a}^{b} r(t)  d t∫ab​r(t) dt  is
i. the total revenue up to time b.
ii. the total revenue from time a to time  bbb .
iii. the change of revenue at any moment.

Accepted Answer

A)   Both ii and iii could be correct.
B)   Only i is correct.
C)   Only iii is correct.
D)   Only ii is correct.

Question 9

Use the definite integral to find the area between the  xxx -axis and the graph of  f(x) f(x) f(x)   over the indicated interval.
- f(x) =x2−6x+9;[2,4]f(x) =x^{2}-6 x+9 ;[2,4]f(x) =x2−6x+9;[2,4]

Accepted Answer

A)    13\frac{1}{3}31​ 
B)    73\frac{7}{3}37​ 
C)    23\frac{2}{3}32​ 
D)    43\frac{4}{3}34​

Question 10

Solve the problem.
-Find the producer's surplus if the supply function is given by  S(x) =x2+4x+20S(x) =x^{2}+4 x+20S(x) =x2+4x+20 . Assume that supply and demand are in equilibrium at  x=24x=24x=24 .

Accepted Answer

A)   10,638
B)   10,368
C)   10,386
D)   10,836

Question 11

Solve the problem.
-Newton's law of cooling states that the rate at which a body changes temperature is proportional to the difference between its temperature and that of the surrounding medium. If a body is in air of temperature  25∘25^{\circ}25∘  and the body cools from  110∘110^{\circ}110∘  to  65∘65^{\circ}65∘  in 30 minutes, find the temperature of the body after 60 minutes. (Round to nearest degree.)

Accepted Answer

A)    47∘47^{\circ}47∘ 
B)    39∘39^{\circ}39∘ 
C)    44∘44^{\circ}44∘ 
D)    40∘40^{\circ}40∘

Question 12

Use the definite integral to find the area between the  xxx -axis and the graph of  f(x) f(x) f(x)   over the indicated interval.
- f(x) =x2(x−2) 2;[0,2]f(x) =x^{2}(x-2) ^{2} ;[0,2]f(x) =x2(x−2) 2;[0,2]

Accepted Answer

A)    1516\frac{15}{16}1615​ 
B)    1615\frac{16}{15}1516​ 
C)    1715\frac{17}{15}1517​ 
D)    1517\frac{15}{17}1715​

Question 13

Approximate the area under the curve and above the  xxx -axis using  nnn  rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle.
- f(x) =x2f(x) =x^{2}f(x) =x2  from  x=0x=0x=0  to  x=4;n=4x=4 ; n=4x=4;n=4

Accepted Answer

A)   36
B)   33
C)   27
D)   30

Question 14

Find the integral.
- ∫(4x+2ex) dx\int\left(\frac{4}{x}+2 e^{x}\right)  d x∫(x4​+2ex) dx

Accepted Answer

A)    4ln⁡∣x∣+2ex+C4 \ln |x|+2 e^{x}+C4ln∣x∣+2ex+C 
B)    4ln⁡∣x∣+2xex−1+C4 \ln |x|+2 x e^{x-1}+C4ln∣x∣+2xex−1+C 
C)    8x2+2xex−1+C\frac{8}{x^{2}}+2 x e^{x-1}+Cx28​+2xex−1+C 
D)    8x2+2ex+C\frac{8}{x^{2}}+2 e^{x}+Cx28​+2ex+C

Question 15

Find the area bounded by the given curves.
- y=x2−5x+4,y=−(x−1) 2y=x^{2}-5 x+4, y=-(x-1) ^{2}y=x2−5x+4,y=−(x−1) 2

Accepted Answer

A)    78\frac{7}{8}87​ 
B)    87\frac{8}{7}78​ 
C)    89\frac{8}{9}98​ 
D)    98\frac{9}{8}89​

Question 16

Find the given indefinite integral.
- ∫ln⁡(7−2x) dx\int \ln (7-2 x)  d x∫ln(7−2x) dx

Accepted Answer

A)    12[(7−2x) ln⁡(7−2x) −14x−1]+C\frac{1}{2}[(7-2 x)  \ln (7-2 x) -14 x-1]+C21​[(7−2x) ln(7−2x) −14x−1]+C 
B)    −12[(7−2x) ln⁡(7−2x) −(7−2x) ]+C-\frac{1}{2}[(7-2 x)  \ln (7-2 x) -(7-2 x) ]+C−21​[(7−2x) ln(7−2x) −(7−2x) ]+C 
C)    17[ln⁡7−2x) −(7−2x) ]+C\left.\frac{1}{7}[\ln 7-2 x) -(7-2 x) \right]+C71​[ln7−2x) −(7−2x) ]+C 
D)    −17[(7−2x) ln⁡(7−2x) −(7−2x) ]+C-\frac{1}{7}[(7-2 x)  \ln (7-2 x) -(7-2 x) ]+C−71​[(7−2x) ln(7−2x) −(7−2x) ]+C

Question 17

Solve the problem.
-Suppose the supply function of a certain item is given by  S(x) =4x+2S(x) =4 x+2S(x) =4x+2  and the demand function is  D(x) =14−x2D(x) =14-x^{2}D(x) =14−x2 . Find the producer's surplus.

Accepted Answer

A)   8
B)    83\frac{8}{3}38​ 
C)   16
D)    163\frac{16}{3}316​

Question 18

Find the area of the shaded region.
-

Accepted Answer

A)   1.39
B)   1.50
C)   1.25
D)   1.69

Question 19

Find the integral.
- ∫(1−6x) e(3x−9x2) dx\int(1-6 x)  e^{\left(3 x-9 x^{2}\right) } d x∫(1−6x) e(3x−9x2) dx

Accepted Answer

A)    13e(3x−9x2) +C\frac{1}{3} e^{\left(3 x-9 x^{2}\right) }+C31​e(3x−9x2) +C 
B)    3(1−6x) e(3x−9x2) +C3(1-6 x)  e^{\left(3 x-9 x^{2}\right) }+C3(1−6x) e(3x−9x2) +C 
C)    13(1−6x) e(3x−9x2) +C\frac{1}{3}(1-6 x)  e^{\left(3 x-9 x^{2}\right) }+C31​(1−6x) e(3x−9x2) +C 
D)    3e(3x−9x2) +C3 e^{\left(3 x-9 x^{2}\right) }+C3e(3x−9x2) +C

Question 20

Solve the problem.
-The rate of growth of profit (in millions)  from an invention is approximated by  P′(x) =xe−x2P^{\prime}(x) =x e^{-x^{2}}P′(x) =xe−x2 , where  xxx  represents time in years. The total profit in year 1 that the invention is in operation is  $\$$  45,000 . Find the total profit function  P(x) \mathrm{P}(\mathrm{x}) P(x)  .

Accepted Answer

A)    P(x) =−12e−x2−229,000P(x) =-\frac{1}{2} e^{-x^{2}}-229,000P(x) =−21​e−x2−229,000 
B)    P(x) =−.5e−x2+0.229P(x) =-.5 e^{-x^{2}}+0.229P(x) =−.5e−x2+0.229 
C)    P(x) =−.5e−x2+229,000P(x) =-.5 e^{-x^{2}}+229,000P(x) =−.5e−x2+229,000 
D)    P(x) =−12e−x2−0.229P(x) =-\frac{1}{2} e^{-x^{2}}-0.229P(x) =−21​e−x2−0.229

Exam 13: Integral Calculus

Approximate the area under the curve and above the $x$ -axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f(x) =x^{2}$ from $x=0$ to $x=4 ; n=4$

Correct Answer
verified

Find the area of the shaded region. -

Correct Answer
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Solve the problem. -Suppose the supply function of a certain item is given by $S(x) =2 x+7$ and the demand function is $D(x) =27-x^{2 / 3}$ . Find the producer's surplus.

Correct Answer
verified

Find the area bounded by the given curves. - $y=x^{2}-5 x+4, y=-(x-1) ^{2}$

Correct Answer
verified

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x) =2 x+7 ;[1,5]$

Correct Answer
verified

Find the integral. - $\int \frac{8}{(y-9) ^{3}} d y$

Correct Answer
verified

Correct Answer
verified

Find the particular solution of the differential equation. - $\frac{d y}{d x}-x y-x=0 ; y=10$ when $x=1$

Correct Answer
verified

Provide an appropriate response. -If $r(t)$ is the rate of change of revenue, then $\int_{a}^{b} r(t) d t$ is i. the total revenue up to time b. ii. the total revenue from time a to time $b$ . iii. the change of revenue at any moment.

Correct Answer
verified

Correct Answer
verified

Find the integral. - $\int\left(\frac{4}{x}+2 e^{x}\right) d x$

Correct Answer
verified

Solve the problem. -Suppose the supply function of a certain item is given by $S(x) =4 x+2$ and the demand function is $D(x) =14-x^{2}$ . Find the producer's surplus.

Correct Answer
verified

Correct Answer
verified

Solve the problem. -The rate of infection of a disease (in people per month) is given by the function $f^{\prime}(t) =\frac{900 t}{t^{2}+1}$ , where $t$ is the time in months since the disease broke out. Find the total number of infected people over the first 5 months of the disease.

Correct Answer
verified

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x) =x^{2}-6 x+9 ;[2,4]$

Correct Answer
verified

Solve the problem. -Find the producer's surplus if the supply function is given by $S(x) =x^{2}+4 x+20$ . Assume that supply and demand are in equilibrium at $x=24$ .

Correct Answer
verified

Find the integral. - $\int(1-6 x) e^{\left(3 x-9 x^{2}\right) } d x$

Correct Answer
verified

Find the given indefinite integral. - $\int \ln (7-2 x) d x$

Correct Answer
verified

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x) =x^{2}(x-2) ^{2} ;[0,2]$

Correct Answer
verified

Find the integral. - $\int 2 e^{4 z} d z$

Correct Answer
verified

Exam 13: Integral Calculus

Approximate the area under the curve and above the xxx -axis using nnn rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - f(x) =x2f(x) =x^{2}f(x) =x2 from x=0x=0x=0 to x=4;n=4x=4 ; n=4x=4;n=4

Correct AnswerverifiedShow Answer

Find the area of the shaded region. -

Correct AnswerverifiedShow Answer

Solve the problem. -Suppose the supply function of a certain item is given by S(x) =2x+7S(x) =2 x+7S(x) =2x+7 and the demand function is D(x) =27−x2/3D(x) =27-x^{2 / 3}D(x) =27−x2/3 . Find the producer's surplus.

Correct AnswerverifiedShow Answer

Find the area bounded by the given curves. - y=x2−5x+4,y=−(x−1) 2y=x^{2}-5 x+4, y=-(x-1) ^{2}y=x2−5x+4,y=−(x−1) 2

Correct AnswerverifiedShow Answer

Use the definite integral to find the area between the xxx -axis and the graph of f(x) f(x) f(x) over the indicated interval. - f(x) =2x+7;[1,5]f(x) =2 x+7 ;[1,5]f(x) =2x+7;[1,5]

Correct AnswerverifiedShow Answer

Find the integral. - ∫8(y−9) 3dy\int \frac{8}{(y-9) ^{3}} d y∫(y−9) 38​dy

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the particular solution of the differential equation. - dydx−xy−x=0;y=10\frac{d y}{d x}-x y-x=0 ; y=10dxdy​−xy−x=0;y=10 when x=1x=1x=1

Correct AnswerverifiedShow Answer

Provide an appropriate response. -If r(t) r(t) r(t) is the rate of change of revenue, then ∫abr(t) dt\int_{a}^{b} r(t) d t∫ab​r(t) dt is i. the total revenue up to time b. ii. the total revenue from time a to time bbb . iii. the change of revenue at any moment.

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the integral. - ∫(4x+2ex) dx\int\left(\frac{4}{x}+2 e^{x}\right) d x∫(x4​+2ex) dx

Correct AnswerverifiedShow Answer

Solve the problem. -Suppose the supply function of a certain item is given by S(x) =4x+2S(x) =4 x+2S(x) =4x+2 and the demand function is D(x) =14−x2D(x) =14-x^{2}D(x) =14−x2 . Find the producer's surplus.

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Use the definite integral to find the area between the xxx -axis and the graph of f(x) f(x) f(x) over the indicated interval. - f(x) =x2−6x+9;[2,4]f(x) =x^{2}-6 x+9 ;[2,4]f(x) =x2−6x+9;[2,4]

Correct AnswerverifiedShow Answer

Solve the problem. -Find the producer's surplus if the supply function is given by S(x) =x2+4x+20S(x) =x^{2}+4 x+20S(x) =x2+4x+20 . Assume that supply and demand are in equilibrium at x=24x=24x=24 .

Correct AnswerverifiedShow Answer

Find the integral. - ∫(1−6x) e(3x−9x2) dx\int(1-6 x) e^{\left(3 x-9 x^{2}\right) } d x∫(1−6x) e(3x−9x2) dx

Correct AnswerverifiedShow Answer

Find the given indefinite integral. - ∫ln⁡(7−2x) dx\int \ln (7-2 x) d x∫ln(7−2x) dx

Correct AnswerverifiedShow Answer

Use the definite integral to find the area between the xxx -axis and the graph of f(x) f(x) f(x) over the indicated interval. - f(x) =x2(x−2) 2;[0,2]f(x) =x^{2}(x-2) ^{2} ;[0,2]f(x) =x2(x−2) 2;[0,2]

Correct AnswerverifiedShow Answer

Find the integral. - ∫2e4zdz\int 2 e^{4 z} d z∫2e4zdz

Correct AnswerverifiedShow Answer

Approximate the area under the curve and above the $x$ -axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f(x) =x^{2}$ from $x=0$ to $x=4 ; n=4$

Correct Answer
verified

Correct Answer
verified

Solve the problem. -Suppose the supply function of a certain item is given by $S(x) =2 x+7$ and the demand function is $D(x) =27-x^{2 / 3}$ . Find the producer's surplus.

Correct Answer
verified

Find the area bounded by the given curves. - $y=x^{2}-5 x+4, y=-(x-1) ^{2}$

Correct Answer
verified

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x) =2 x+7 ;[1,5]$

Correct Answer
verified

Find the integral. - $\int \frac{8}{(y-9) ^{3}} d y$

Correct Answer
verified

Correct Answer
verified

Find the particular solution of the differential equation. - $\frac{d y}{d x}-x y-x=0 ; y=10$ when $x=1$

Correct Answer
verified

Provide an appropriate response. -If $r(t)$ is the rate of change of revenue, then $\int_{a}^{b} r(t) d t$ is i. the total revenue up to time b. ii. the total revenue from time a to time $b$ . iii. the change of revenue at any moment.

Correct Answer
verified

Correct Answer
verified

Find the integral. - $\int\left(\frac{4}{x}+2 e^{x}\right) d x$

Correct Answer
verified

Solve the problem. -Suppose the supply function of a certain item is given by $S(x) =4 x+2$ and the demand function is $D(x) =14-x^{2}$ . Find the producer's surplus.

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x) =x^{2}-6 x+9 ;[2,4]$

Correct Answer
verified

Solve the problem. -Find the producer's surplus if the supply function is given by $S(x) =x^{2}+4 x+20$ . Assume that supply and demand are in equilibrium at $x=24$ .

Correct Answer
verified

Find the integral. - $\int(1-6 x) e^{\left(3 x-9 x^{2}\right) } d x$

Correct Answer
verified

Find the given indefinite integral. - $\int \ln (7-2 x) d x$

Correct Answer
verified

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x) =x^{2}(x-2) ^{2} ;[0,2]$

Correct Answer
verified

Find the integral. - $\int 2 e^{4 z} d z$

Correct Answer
verified