Question 1

A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 2 t$ . Find the instantaneous velocity of the particle at time t = 1.

Accepted Answer

A) 1 
B) 4 
C) 0 
D) 3 
E) 8
F)6
G)5
H)2 
A) 1 
B) 4 
C) 0 
D) 3 
E) 8
F)6
G)5
H)2

Question 2

The cost function of manufacturing x meters of a fabric is C (x) = 25,000 + 3x - $0.002 x ^ { 2 }$ + $0.000001 x ^ { 3 }$ . Find $\mathrm { C } ^ { ' }$ (5000).

Accepted Answer

A) 58 
B) 580 
C) 5.8 
D) 5800 
E) 60
F)600
G)6
H)6000 
A) 58 
B) 580 
C) 5.8 
D) 5800 
E) 60
F)600
G)6
H)6000

Question 3

Show that the rate of change of the circumference of a circle, with respect to the radius of the circle, does not depend on the radius.

Accepted Answer

The answer of Show that the rate of change of...

Question 4

Find the slope of the tangent to the curve $x y = 8$ at the point $( - 2,4 )$ .

Accepted Answer

A)  $- 4$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)4
H)8 
A)  $- 4$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)4
H)8

Question 5

Let $f ( x ) = \ln \left( \sin ^ { 2 } x + 1 \right)$ . Find the value of $f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

Accepted Answer

A) 0 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 2 } { 3 }$ 
D) 1 
E)  $\frac { 3 } { 2 }$ 
F)
$\sqrt { 2 }$
G)2
H)3 
A) 0 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 2 } { 3 }$ 
D) 1 
E)  $\frac { 3 } { 2 }$ 
F)
$\sqrt { 2 }$
G)2
H)3

Question 6

Show that the curve $y = x ^ { 5 } + 2 x ^ { 3 } + 4 x$ has no tangent line with slope 3.

Accepted Answer

The answer of Show that the curve \(y = x...

Question 7

If $g ( x ) = \sin 2 x , \text { find the value of } g ^ { ( 6 ) } ( 0 )$

Accepted Answer

A) 0 
B) 2 
C)  $- 2$ 
D) 8 
E)  $- 8$ 
F)64
G) $- 64$

H)128 
A) 0 
B) 2 
C)  $- 2$ 
D) 8 
E)  $- 8$ 
F)64
G) $- 64$

H)128

Question 8

If $x \cos y + y \cos x = 1$ , find an expression for $\frac { d y } { d x }$ .

Accepted Answer

The answer of If \(x \cos y + y \cos...

Question 9

Find $\frac { d y } { d x }$ implicitly:
(a) $y = ( \sin x ) ^ { \cos x }$ (b) $y \ln x = \sin \left( y ^ { 2 } \right)$

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 10

Find the derivative of $f ( x ) = \cos x - \sin x$

Accepted Answer

A)  $\sin x + \cos x$ 
B)  $- \sin x + \cos x$ 
C)  $\sin x - \cos x$ 
D)  $- \sin x - \cos x$ 
E)  $2 \sin x$
F) $2 \cos x$
G) $- 2 \sin x$

H) $- 2 \cos x$ 
A)  $\sin x + \cos x$ 
B)  $- \sin x + \cos x$ 
C)  $\sin x - \cos x$ 
D)  $- \sin x - \cos x$ 
E)  $2 \sin x$
F) $2 \cos x$
G) $- 2 \sin x$

H) $- 2 \cos x$

Question 11

Let $p ( t ) = \frac { 20,000 } { 1 + 1999 e ^ { - 0.1 t } }$ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

Accepted Answer

The answer of Let \(p ( t ) = \frac...

Question 12

If sin y = cos x, find the value of $\frac { d y } { d x }$ at the point $\left( \frac { 4 \pi } { 3 } , - \frac { \pi } { 6 } \right)$ .

Accepted Answer

The answer of If sin y = cos x, find...

Question 13

Find the slope of the normal line to the curve $y = 4 x ^ { 2 } - 5 x$ at the point (1, $- 1 )$ .

Accepted Answer

A) 8 
B)  $- 5$ 
C) 3 
D) 1 
E)  $- \frac { 1 } { 3 }$
F) $\frac { 1 } { 5 }$
G) $- \frac { 1 } { 8 }$ 
H)0 
A) 8 
B)  $- 5$ 
C) 3 
D) 1 
E)  $- \frac { 1 } { 3 }$
F) $\frac { 1 } { 5 }$
G) $- \frac { 1 } { 8 }$ 
H)0

Question 14

The curve y = x $3$ +x $2$ - x has two horizontal tangents. Find the distance between these two horizontal lines.

Accepted Answer

A)  $\frac { 11 } { 9 }$ 
B)  $\frac { 22 } { 27 }$ 
C)  $\frac { 32 } { 27 }$ 
D)  $\frac { 5 } { 3 }$ 
E)  $\frac { 14 } { 9 }$ 
F) $\frac { 4 } { 3 }$ 
G) $\frac { 13 } { 9 }$

H) $\frac { 7 } { 3 }$ 
A)  $\frac { 11 } { 9 }$ 
B)  $\frac { 22 } { 27 }$ 
C)  $\frac { 32 } { 27 }$ 
D)  $\frac { 5 } { 3 }$ 
E)  $\frac { 14 } { 9 }$ 
F) $\frac { 4 } { 3 }$ 
G) $\frac { 13 } { 9 }$

H) $\frac { 7 } { 3 }$

Question 15

Given $x = e ^ { t } , y = \sin t$ find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ when $t = 0$

Accepted Answer

A)  $- \frac { 1 } { \sqrt { 2 } }$ 
B)  $- 2$ 
C)  $\frac { 1 } { \sqrt { 2 } }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $- 1$
F)2
G)1
H) $- \frac { 1 } { 2 }$ 
A)  $- \frac { 1 } { \sqrt { 2 } }$ 
B)  $- 2$ 
C)  $\frac { 1 } { \sqrt { 2 } }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $- 1$
F)2
G)1
H) $- \frac { 1 } { 2 }$

Question 16

Suppose that a baseball is tossed straight upward and that its height (in feet) as a function of time (in seconds) is given by the formula h(t) = 128t - 16t2.
(a) Find the instantaneous velocity and acceleration of the baseball at time t.
(b) What is the maximum height attained by the ball?
(c) What is the average velocity of the ball during the time interval from t = 1 to t = 4?
(d) How long does it take before the ball lands?
(e) At what time is the height of the ball 112 feet?

Accepted Answer

(a) v(t) = @#LAT-DLM& (t) = 128 - 32t ft/s
a(t) =

Question 17

Suppose that $f ( x ) = ( 1 + 2 x ) ^ { 3 }$ , find $f ^ { \prime } ( - 1 )$

Accepted Answer

A) 0 
B) 2 
C) 4 
D) 6 
E) 8
F) $- 2$ 
G) $- 4$ 
H) $- 6$ 
A) 0 
B) 2 
C) 4 
D) 6 
E) 8
F) $- 2$ 
G) $- 4$ 
H) $- 6$

Question 18

Let $f ( x ) = \tan ^ { - 1 } x + \tan ^ { - 1 } \frac { 1 } { x }$ .(a) Show that $f ( x )$ is constant on $( - \infty , 0 ) \text { and } ( 0 , \infty )$ .(b) Determine the value of the constant(s).

Accepted Answer

(a) @#LAT-DLM& Since @#LAT-DLM& (x) = 0 for all x in the dom

Question 19

If $f ( x ) = ( x + 1 ) ^ { 2 } ( x + 2 ) ^ { 3 } , \text { find } f ^ { \prime } ( 0 )$

Accepted Answer

A) 4 
B) 24 
C) 28 
D) 12 
E) 16
F)8
G)32
H)6 
A) 4 
B) 24 
C) 28 
D) 12 
E) 16
F)8
G)32
H)6

Question 20

Let $f ( x ) = x \ln \left( x ^ { 2 } - 3 \right)$ . Find the value of $f ^ { \prime } ( 2 )$ .

Accepted Answer

A) 0 
B) 2 
C) 4 
D) 6 
E) 8
F)10
G)12
H)14 
A) 0 
B) 2 
C) 4 
D) 6 
E) 8
F)10
G)12
H)14

A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 2 t$ . Find the instantaneous velocity of the particle at time t = 1.

The cost function of manufacturing x meters of a fabric is C (x) = 25,000 + 3x - $0.002 x ^ { 2 }$ + $0.000001 x ^ { 3 }$ . Find $\mathrm { C } ^ { ' }$ (5000).

Show that the rate of change of the circumference of a circle, with respect to the radius of the circle, does not depend on the radius.

Find the slope of the tangent to the curve $x y = 8$ at the point $( - 2,4 )$ .

Let $f ( x ) = \ln \left( \sin ^ { 2 } x + 1 \right)$ . Find the value of $f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

Show that the curve $y = x ^ { 5 } + 2 x ^ { 3 } + 4 x$ has no tangent line with slope 3.

If $g ( x ) = \sin 2 x , \text { find the value of } g ^ { ( 6 ) } ( 0 )$

If $x \cos y + y \cos x = 1$ , find an expression for $\frac { d y } { d x }$ .

Find $\frac { d y } { d x }$ implicitly: (a) $y = ( \sin x ) ^ { \cos x }$ (b) $y \ln x = \sin \left( y ^ { 2 } \right)$

Find the derivative of $f ( x ) = \cos x - \sin x$

Let $p ( t ) = \frac { 20,000 } { 1 + 1999 e ^ { - 0.1 t } }$ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

If sin y = cos x, find the value of $\frac { d y } { d x }$ at the point $\left( \frac { 4 \pi } { 3 } , - \frac { \pi } { 6 } \right)$ .

Find the slope of the normal line to the curve $y = 4 x ^ { 2 } - 5 x$ at the point (1, $- 1 )$ .

The curve y = x $3$ +x $2$ - x has two horizontal tangents. Find the distance between these two horizontal lines.

Given $x = e ^ { t } , y = \sin t$ find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ when $t = 0$

Suppose that $f ( x ) = ( 1 + 2 x ) ^ { 3 }$ , find $f ^ { \prime } ( - 1 )$

Let $f ( x ) = \tan ^ { - 1 } x + \tan ^ { - 1 } \frac { 1 } { x }$ .(a) Show that $f ( x )$ is constant on $( - \infty , 0 ) \text { and } ( 0 , \infty )$ .(b) Determine the value of the constant(s).

If $f ( x ) = ( x + 1 ) ^ { 2 } ( x + 2 ) ^ { 3 } , \text { find } f ^ { \prime } ( 0 )$

Let $f ( x ) = x \ln \left( x ^ { 2 } - 3 \right)$ . Find the value of $f ^ { \prime } ( 2 )$ .

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Exam 3: Differentiation Rules

A particle moves along a straight line with equation of motion s=t2−2ts = t ^ { 2 } - 2 ts=t2−2t . Find the instantaneous velocity of the particle at time t = 1.

The cost function of manufacturing x meters of a fabric is C (x) = 25,000 + 3x - 0.002x20.002 x ^ { 2 }0.002x2 + 0.000001x30.000001 x ^ { 3 }0.000001x3 . Find C′\mathrm { C } ^ { ' }C′ (5000).

Show that the rate of change of the circumference of a circle, with respect to the radius of the circle, does not depend on the radius.

Find the slope of the tangent to the curve xy=8x y = 8xy=8 at the point (−2,4)( - 2,4 )(−2,4) .

Let f(x)=ln⁡(sin⁡2x+1)f ( x ) = \ln \left( \sin ^ { 2 } x + 1 \right)f(x)=ln(sin2x+1) . Find the value of f′(π4)f ^ { \prime } \left( \frac { \pi } { 4 } \right)f′(4π​) .

Show that the curve y=x5+2x3+4xy = x ^ { 5 } + 2 x ^ { 3 } + 4 xy=x5+2x3+4x has no tangent line with slope 3.

If g(x)=sin⁡2x, find the value of g(6)(0)g ( x ) = \sin 2 x , \text { find the value of } g ^ { ( 6 ) } ( 0 )g(x)=sin2x, find the value of g(6)(0)

If xcos⁡y+ycos⁡x=1x \cos y + y \cos x = 1xcosy+ycosx=1 , find an expression for dydx\frac { d y } { d x }dxdy​ .

Find dydx\frac { d y } { d x }dxdy​ implicitly: (a) y=(sin⁡x)cos⁡xy = ( \sin x ) ^ { \cos x }y=(sinx)cosx (b) yln⁡x=sin⁡(y2)y \ln x = \sin \left( y ^ { 2 } \right)ylnx=sin(y2)

Find the derivative of f(x)=cos⁡x−sin⁡xf ( x ) = \cos x - \sin xf(x)=cosx−sinx

Let p(t)=20,0001+1999e−0.1tp ( t ) = \frac { 20,000 } { 1 + 1999 e ^ { - 0.1 t } }p(t)=1+1999e−0.1t20,000​ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

If sin y = cos x, find the value of dydx\frac { d y } { d x }dxdy​ at the point (4π3,−π6)\left( \frac { 4 \pi } { 3 } , - \frac { \pi } { 6 } \right)(34π​,−6π​) .

Find the slope of the normal line to the curve y=4x2−5xy = 4 x ^ { 2 } - 5 xy=4x2−5x at the point (1, −1)- 1 )−1) .

The curve y = x 333 +x 222 - x has two horizontal tangents. Find the distance between these two horizontal lines.

Given x=et,y=sin⁡tx = e ^ { t } , y = \sin tx=et,y=sint find the value of d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ when t=0t = 0t=0

Suppose that f(x)=(1+2x)3f ( x ) = ( 1 + 2 x ) ^ { 3 }f(x)=(1+2x)3 , find f′(−1)f ^ { \prime } ( - 1 )f′(−1)

If f(x)=(x+1)2(x+2)3, find f′(0)f ( x ) = ( x + 1 ) ^ { 2 } ( x + 2 ) ^ { 3 } , \text { find } f ^ { \prime } ( 0 )f(x)=(x+1)2(x+2)3, find f′(0)

Let f(x)=xln⁡(x2−3)f ( x ) = x \ln \left( x ^ { 2 } - 3 \right)f(x)=xln(x2−3) . Find the value of f′(2)f ^ { \prime } ( 2 )f′(2) .

Functions and Models

Limits and Derivatives

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Vector Calculus

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A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 2 t$ . Find the instantaneous velocity of the particle at time t = 1.

The cost function of manufacturing x meters of a fabric is C (x) = 25,000 + 3x - $0.002 x ^ { 2 }$ + $0.000001 x ^ { 3 }$ . Find $\mathrm { C } ^ { ' }$ (5000).

Find the slope of the tangent to the curve $x y = 8$ at the point $( - 2,4 )$ .

Let $f ( x ) = \ln \left( \sin ^ { 2 } x + 1 \right)$ . Find the value of $f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

Show that the curve $y = x ^ { 5 } + 2 x ^ { 3 } + 4 x$ has no tangent line with slope 3.

If $g ( x ) = \sin 2 x , \text { find the value of } g ^ { ( 6 ) } ( 0 )$

If $x \cos y + y \cos x = 1$ , find an expression for $\frac { d y } { d x }$ .

Find $\frac { d y } { d x }$ implicitly: (a) $y = ( \sin x ) ^ { \cos x }$ (b) $y \ln x = \sin \left( y ^ { 2 } \right)$

Find the derivative of $f ( x ) = \cos x - \sin x$

Let $p ( t ) = \frac { 20,000 } { 1 + 1999 e ^ { - 0.1 t } }$ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

If sin y = cos x, find the value of $\frac { d y } { d x }$ at the point $\left( \frac { 4 \pi } { 3 } , - \frac { \pi } { 6 } \right)$ .

Find the slope of the normal line to the curve $y = 4 x ^ { 2 } - 5 x$ at the point (1, $- 1 )$ .

The curve y = x $3$ +x $2$ - x has two horizontal tangents. Find the distance between these two horizontal lines.

Given $x = e ^ { t } , y = \sin t$ find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ when $t = 0$

Suppose that $f ( x ) = ( 1 + 2 x ) ^ { 3 }$ , find $f ^ { \prime } ( - 1 )$

If $f ( x ) = ( x + 1 ) ^ { 2 } ( x + 2 ) ^ { 3 } , \text { find } f ^ { \prime } ( 0 )$

Let $f ( x ) = x \ln \left( x ^ { 2 } - 3 \right)$ . Find the value of $f ^ { \prime } ( 2 )$ .