Question 1

Find the derivative of the function.  f(x) f(x) f(x)   = sinh 4x

Accepted Answer

A)   -4 cosh 4x
B)   4 sinh 4x
C)   -sinh 4x
D)   4 cosh 4x E) None of the above
F) C) and D)

Question 2

If  y=3x2+6xy=3 x^{2}+6 xy=3x2+6x  and  dxdt=5\frac{d x}{d t}=5dtdx​=5  , find  dydt\frac{d y}{d t}dtdy​  when  x=2x=2x=2  .

Accepted Answer

A)   60
B)    210210210 
C)   300
D)   260
E)   None of these F) B) and E)
G) None of the above

Question 3

Differentiate the function.  g(t) =t5ln⁡9tg(t) =t^{5} \ln 9 tg(t) =t5ln9t

Accepted Answer

A)    1+ln⁡9t9t1+\frac{\ln 9 t}{9 t}1+9tln9t​ 
B)    59t3\frac{5}{9} t^{3}95​t3 
C)    t4(1+5ln⁡9t) t^{4}(1+5 \ln 9 t) t4(1+5ln9t)  
D)    t4(19+5ln⁡9t) t^{4}\left(\frac{1}{9}+5 \ln 9 t\right) t4(91​+5ln9t) E) A) and B)
F) None of the above

Question 4

Find the derivative of the function.  y=8cos⁡−1(sin⁡−1t) y=8 \cos ^{-1}\left(\sin ^{-1} t\right) y=8cos−1(sin−1t)

Accepted Answer

A)    y′=−81−(sin⁡−1(t) ) 2y^{\prime}=-\frac{8}{\sqrt{1-\left(\sin ^{-1}(t) \right) ^{2}}}y′=−1−(sin−1(t) ) 2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​8​ 
B)    y′=−8(1−t2) (1−sin⁡−1(t) ) y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right) \left(1-\sin ^{-1}(t) \right) }}y′=−(1−t2) (1−sin−1(t) ) c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​8​ 
C)    y′=−8(1+t2) (1+(sin⁡−1(t) ) 2) y^{\prime}=-\frac{8}{\sqrt{\left(1+t^{2}\right) \left(1+\left(\sin ^{-1}(t) \right) ^{2}\right) }}y′=−(1+t2) (1+(sin−1(t) ) 2) c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z'/>​8​ 
D)    y′=−8(1−t2) (1−(sin⁡−1(t) ) 2) y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right) \left(1-\left(\sin ^{-1}(t) \right) ^{2}\right) }}y′=−(1−t2) (1−(sin−1(t) ) 2) c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z'/>​8​ 
E)    y′=−8(1−t2) y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right) }}y′=−(1−t2) c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​8​F) A) and D)
G) C) and D)

Question 5

Use logarithmic differentiation to find the derivative of the function.  y=x8+1x−13y=\sqrt[3]{\frac{x^{8}+1}{x-1}}y=3x−1x8+1​​

Accepted Answer

A)    7x8−8x7+13(x8+1) 2/3(x−1) 4/3\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right) ^{2 / 3}(x-1) ^{4 / 3}}3(x8+1) 2/3(x−1) 4/37x8−8x7+1​ 
B)    7x8−8x7−13(x8+1) 2/3(x−1) 4/3\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right) ^{2 / 3}(x-1) ^{4 / 3}}3(x8+1) 2/3(x−1) 4/37x8−8x7−1​ 
C)    7x8−8x7+13(x8+1) (x−1) \frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right) (x-1) }3(x8+1) (x−1) 7x8−8x7+1​ 
D)    7x8−8x7−13(x8+1) (x−1) \frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right) (x-1) }3(x8+1) (x−1) 7x8−8x7−1​E) All of the above
F) A) and D)

Question 6

Suppose that  F(x) =f(g(x) ) F(x) =f(g(x) ) F(x) =f(g(x) )   and  g(14) =2g(14) =2g(14) =2  ,  g′(14) =5g^{\prime}(14) =5g′(14) =5  ,  f′(14) =15f^{\prime}(14) =15f′(14) =15  , and  f′(2) =16f^{\prime}(2) =16f′(2) =16  . Find  F′(14) F^{\prime}(14) F′(14)   .

Accepted Answer

A)   20
B)    808080 
C)   24
D)   140
E)   17 F) None of the above
G) A) and C)

Question 7

Two cars start moving from the same point. One travels south at  505050  mi/h and the other travels west at  404040  mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Accepted Answer

A)    55.42 mi/h55.42 ~\mathrm{mi} / \mathrm{h}55.42 mi/h 
B)    76.43 mi/h76.43 ~\mathrm{mi} / \mathrm{h}76.43 mi/h 
C)    81.38 mi/h81.38 ~\mathrm{mi} / \mathrm{h}81.38 mi/h 
D)    65.49 mi/h65.49 ~\mathrm{mi} / \mathrm{h}65.49 mi/h 
E)    64.03 mi/h64.03~ \mathrm{mi} / \mathrm{h}64.03 mi/hF) C) and E)
G) B) and E)

Question 8

Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$

Accepted Answer

The answer of Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$...

Question 9

Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$

Accepted Answer

The answer of Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$...

Question 10

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=t^{10} e^{-t}$ ; t = 1

Accepted Answer

The answer of s(t) is the position of a body...

Question 11

Find the derivative of the function.  h(t) =t9−3t7+8t6et8t6h(t) =\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}h(t) =8t6t9−3t7+8t6et​

Accepted Answer

A)    3t2−3+8et3 t^{2}-3+8 e^{t}3t2−3+8et 
B)    3t2−3+et3 t^{2}-3+e^{t}3t2−3+et 
C)    38\frac{3}{8}83​   t2−t^{2}-t2−   38\frac{3}{8}83​   +et+e^{t}+et 
D)    9t8−21t6+et9 t^{8}-21 t^{6}+e^{t}9t8−21t6+etE) B) and C)
F) A) and D)

Question 12

Find the average rate of change of the area of a circle with respect to its radius r as r changes from 5 to 6 .

Accepted Answer

A)    11π11 \pi11π 
B)    36π36 \pi36π 
C)    8π8 \pi8π 
D)    6π6 \pi6π 
E)    12π12 \pi12πF) None of the above
G) A) and C)

Question 13

Calculate $y^{\prime}$ . $x e^{y}=y-8$

Accepted Answer

The answer of Calculate $y^{\prime}$ . $x e^{y}=y-8$...

Question 14

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $3,500 \text { dynes } / \mathrm{cm}^{2}$ and viscosity $\eta$ =.028.
Find the velocity of the blood at radius r = $0.001$

Accepted Answer

The answer of Refer to the law of laminar flow....

Question 15

Find the rate of change of y with respect to x at the given values of x and y. $2 x y^{2}-5 x^{2} y-375=0$ ; x = 3, y = -5

Accepted Answer

The answer of Find the rate of change of y...

Question 16

If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as $V=5000\left(1-\frac{t}{40}\right)^{2}$ .
Find the rate at which water is draining from the tank after 9 minutes.

Accepted Answer

The answer of If a tank holds 5000 gallons of...

Question 17

If  f(x) =6cos⁡x+sin⁡2xf(x) =6 \cos x+\sin ^{2} xf(x) =6cosx+sin2x  , find  f′(x) f^{\prime}(x) f′(x)   and  f′′(x) f^{\prime \prime}(x) f′′(x)   .

Accepted Answer

A)    f′′(x) =−6cos⁡(2x) +2cos⁡(x) f^{\prime \prime}(x) =-6 \cos (2 x) +2 \cos (x) f′′(x) =−6cos(2x) +2cos(x)   
B)    f′(x) =−6sin⁡(x) +sin⁡(2x) f^{\prime}(x) =-6 \sin (x) +\sin (2 x) f′(x) =−6sin(x) +sin(2x)   
C)    f′(x) =−6sin⁡(2x) +sin⁡(x) f^{\prime}(x) =-6 \sin (2 x) +\sin (x) f′(x) =−6sin(2x) +sin(x)   
D)    f′′(x) =−6cos⁡(x) +2cos⁡(2x) f^{\prime \prime}(x) =-6 \cos (x) +2 \cos (2 x) f′′(x) =−6cos(x) +2cos(2x)   
E)    f′′(x) =−2cos⁡(2x) +6cos⁡(x) f^{\prime \prime}(x) =-2 \cos (2 x) +6 \cos (x) f′′(x) =−2cos(2x) +6cos(x)   F) B) and C)
G) B) and D)

Question 18

If  f(t) =9t+1f(t) =\sqrt{9 t+1}f(t) =9t+1​  , find  f′′(4) f^{\prime \prime}(4) f′′(4)   .

Accepted Answer

A)    −0.010-0.010−0.010 
B)    0.0150.0150.015 
C)    −0.033-0.033−0.033 
D)    −0.22-0.22−0.22 
E)    0.0440.0440.044F) B) and D)
G) C) and D)

Question 19

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy6;  (π2,1) \left(\frac{\pi}{2}, 1\right) (2π​,1)

Accepted Answer

A)   x =  π2\frac{\pi}{2}2π​ 
B)   y = 6x + 1
C)   y = x
D)   y = 1 E) C) and D)
F) A) and D)

Question 20

Find all points at which the tangent line is horizontal on the graph of the function. $y(x)=6 \sin x+\sin ^{2} x$

Accepted Answer

The answer of Find all points at which the tangent...

Exam 3: Applications of Differentiation

Differentiate the function. g(t) =t5ln⁡9tg(t) =t^{5} \ln 9 tg(t) =t5ln9t

Correct AnswerverifiedShow Answer

Find the rate of change of y with respect to x at the given values of x and y. 2xy2−5x2y−375=02 x y^{2}-5 x^{2} y-375=02xy2−5x2y−375=0 ; x = 3, y = -5

Correct AnswerverifiedShow Answer

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3,500 dynes /cm23,500 \text { dynes } / \mathrm{cm}^{2}3,500 dynes /cm2 and viscosity η\etaη =.028. Find the velocity of the blood at radius r = 0.0010.0010.001

Correct AnswerverifiedShow Answer

Find all points at which the tangent line is horizontal on the graph of the function. y(x)=6sin⁡x+sin⁡2xy(x)=6 \sin x+\sin ^{2} xy(x)=6sinx+sin2x

Correct AnswerverifiedShow Answer

Find f′(a)f^{\prime}(a)f′(a) . f(x)=2+x−5x5f(x)=2+x-5 x^{5}f(x)=2+x−5x5

Correct AnswerverifiedShow Answer

Suppose that F(x) =f(g(x) ) F(x) =f(g(x) ) F(x) =f(g(x) ) and g(14) =2g(14) =2g(14) =2 , g′(14) =5g^{\prime}(14) =5g′(14) =5 , f′(14) =15f^{\prime}(14) =15f′(14) =15 , and f′(2) =16f^{\prime}(2) =16f′(2) =16 . Find F′(14) F^{\prime}(14) F′(14) .

Correct AnswerverifiedShow Answer

Two cars start moving from the same point. One travels south at 505050 mi/h and the other travels west at 404040 mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Correct AnswerverifiedShow Answer

Find the average rate of change of the area of a circle with respect to its radius r as r changes from 5 to 6 .

Correct AnswerverifiedShow Answer

If f(x) =6cos⁡x+sin⁡2xf(x) =6 \cos x+\sin ^{2} xf(x) =6cosx+sin2x , find f′(x) f^{\prime}(x) f′(x) and f′′(x) f^{\prime \prime}(x) f′′(x) .

Correct AnswerverifiedShow Answer

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where t≥0t \geq 0t≥0 . Find the position, velocity, and speed of the body at the indicated time. s(t)=t10e−ts(t)=t^{10} e^{-t}s(t)=t10e−t ; t = 1

Correct AnswerverifiedShow Answer

If f(t) =9t+1f(t) =\sqrt{9 t+1}f(t) =9t+1​ , find f′′(4) f^{\prime \prime}(4) f′′(4) .

Correct AnswerverifiedShow Answer

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy6; (π2,1) \left(\frac{\pi}{2}, 1\right) (2π​,1)

Correct AnswerverifiedShow Answer

Calculate y′y^{\prime}y′ . y=3xcos⁡xy=3 \sqrt{x} \cos \sqrt{x}y=3x​cosx​

Correct AnswerverifiedShow Answer

Find the derivative of the function. h(t) =t9−3t7+8t6et8t6h(t) =\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}h(t) =8t6t9−3t7+8t6et​

Correct AnswerverifiedShow Answer

Find the derivative of the function. f(x) f(x) f(x) = sinh 4x

Correct AnswerverifiedShow Answer

If y=3x2+6xy=3 x^{2}+6 xy=3x2+6x and dxdt=5\frac{d x}{d t}=5dtdx​=5 , find dydt\frac{d y}{d t}dtdy​ when x=2x=2x=2 .

Correct AnswerverifiedShow Answer

Use logarithmic differentiation to find the derivative of the function. y=x8+1x−13y=\sqrt[3]{\frac{x^{8}+1}{x-1}}y=3x−1x8+1​​

Correct AnswerverifiedShow Answer

Find the derivative of the function. y=8cos⁡−1(sin⁡−1t) y=8 \cos ^{-1}\left(\sin ^{-1} t\right) y=8cos−1(sin−1t)

Correct AnswerverifiedShow Answer

Calculate y′y^{\prime}y′ . xey=y−8x e^{y}=y-8xey=y−8

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Differentiate the function. $g(t) =t^{5} \ln 9 t$

Correct Answer
verified

Find the rate of change of y with respect to x at the given values of x and y. $2 x y^{2}-5 x^{2} y-375=0$ ; x = 3, y = -5

Correct Answer
verified

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $3,500 \text { dynes } / \mathrm{cm}^{2}$ and viscosity $\eta$ =.028. Find the velocity of the blood at radius r = $0.001$

Correct Answer
verified

Find all points at which the tangent line is horizontal on the graph of the function. $y(x)=6 \sin x+\sin ^{2} x$

Correct Answer
verified

Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$

Correct Answer
verified

Suppose that $F(x) =f(g(x) )$ and $g(14) =2$ , $g^{\prime}(14) =5$ , $f^{\prime}(14) =15$ , and $f^{\prime}(2) =16$ . Find $F^{\prime}(14)$ .

Correct Answer
verified

Two cars start moving from the same point. One travels south at $50$ mi/h and the other travels west at $40$ mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Correct Answer
verified

Correct Answer
verified

If $f(x) =6 \cos x+\sin ^{2} x$ , find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

Correct Answer
verified

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=t^{10} e^{-t}$ ; t = 1

Correct Answer
verified

If $f(t) =\sqrt{9 t+1}$ , find $f^{\prime \prime}(4)$ .

Correct Answer
verified

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy⁶; $\left(\frac{\pi}{2}, 1\right)$

Correct Answer
verified

Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$

Correct Answer
verified

Find the derivative of the function. $h(t) =\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}$

Correct Answer
verified

Find the derivative of the function. $f(x)$ = sinh 4x

Correct Answer
verified

If $y=3 x^{2}+6 x$ and $\frac{d x}{d t}=5$ , find $\frac{d y}{d t}$ when $x=2$ .

Correct Answer
verified

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[3]{\frac{x^{8}+1}{x-1}}$

Correct Answer
verified

Find the derivative of the function. $y=8 \cos ^{-1}\left(\sin ^{-1} t\right)$

Correct Answer
verified

Calculate $y^{\prime}$ . $x e^{y}=y-8$

Correct Answer
verified

Correct Answer
verified