Question 1

Given $f ( x ) = e ^ { x } - x ^ { 3 }$ , find $f ^ { \prime } ( x ) , f ^ { \prime \prime } ( x )$ and $f ^ { ( 3 ) } ( x )$ .

Accepted Answer

The answer of Given \(f ( x ) = e...

Question 2

$\begin{array} { | r | c | c | c | c | } 
\hline x & f ( x ) & f ^ { \prime } ( x ) & g ( x ) & g ^ { \prime } ( x ) \
\hline - 6 & 7 & - 8 & - 6 & 7 \
- 4 & 1 & - 5 & 0 & 5 \
- 2 & 3 & - 2 & 4 & 3 \
0 & 5 & 0 & 6 & 1 \
2 & - 5 & 1 & 6 & - 1 \
4 & - 3 & 3 & 4 & - 3 \
6 & 1 & 5 & 0 & - 5 \
\hline
\end{array}$
-Find $\frac { d } { d x } \left( x ^ { 2 } f ( x ) \right) \text { when } x = 4 \text {. }$

Accepted Answer

A)  $- 24$ 
B)  $- 18$ 
C) 0 
D) 12 
E) 24
F)36
G)48
H)72 
A)  $- 24$ 
B)  $- 18$ 
C) 0 
D) 12 
E) 24
F)36
G)48
H)72

Question 3

Find the x-coordinate of the point at which the graph of $y = \frac { \ln x } { x }$ has a horizontal tangent.

Accepted Answer

The answer of Find the x-coordinate of the point at...

Question 4

Find the derivatives of the following functions:
(a) $f ( x ) = \sin ^ { - 1 } \left( x ^ { 2 } \right)$ (b) $f ( x ) = \left( \sin ^ { - 1 } x \right) ^ { 2 }$ (c) $f ( x ) = x \sin ^ { - 1 } \left( x ^ { 3 } \right)$ (d) $f ( x ) = \tan ^ { - 1 } \left( \frac { x + 1 } { x - 1 } \right)$ (e) $f ( x ) = \cos ^ { - 1 } \left( e ^ { x } \right)$

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 5

Find the equation of the line normal to the curve defined by the equation $x ^ { 3 } y ^ { 4 } - 5 = x ^ { 3 } - x ^ { 2 } + y$ at the point (2, $- 1$ ).

Accepted Answer

A)  $33 x + 4 y - 62 = 0$ 
B)  $33 x - 4 y - 62 = 0$ 
C)  $33 x + 4 y + 62 = 0$ 
D)  $33 x - 4 y + 62 = 0$ 
E)  $4 x - 33 y - 41 = 0$
F) $4 x + 33 y - 41 = 0$
G) $4 x - 33 y + 41 = 0$
H) $4 x + 33 y + 41 = 0$ 
A)  $33 x + 4 y - 62 = 0$ 
B)  $33 x - 4 y - 62 = 0$ 
C)  $33 x + 4 y + 62 = 0$ 
D)  $33 x - 4 y + 62 = 0$ 
E)  $4 x - 33 y - 41 = 0$
F) $4 x + 33 y - 41 = 0$
G) $4 x - 33 y + 41 = 0$
H) $4 x + 33 y + 41 = 0$

Question 6

Find the point(s) where the curve $2 x ^ { 2 } - 2 x y + y ^ { 2 } = 4$ has a horizontal tangent.

Accepted Answer

The answer of Find the point(s) where the curve \(2...

Question 7

Differentiate each of the following functions:
(a) $f ( x ) = x ^ { 3 } \cos x$ (b) $f ( x ) = \frac { x } { \sin x }$ (c) $f ( x ) = \frac { x \sin x } { x + \sin x }$

Accepted Answer

The answer of Differentiate each of the following functions:
(a) \(f...

Question 8

A physics experiment involving the acceleration of shuttle on a rail produced the following
data: $$\begin{array} { | c | c | c | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Time (s) } & 0.017 & 0.05 & 0.083 & 0.117 & 0.15 & 0.183 & 0.217 & 0.25 & 0.283 & 0.317 & 0.35 & 0.383 \
\hline \text { Position (m) } & 0.020 & 0.046 & 0.081 & 0.129 & 0.187 & 0.256 & 0.337 & 0.427 & 0.529 & 0.640 & 0.765 & 0.899 \
\hline
\end{array}$$ (a) Make a scatter plot of the data.(b) Fit a quadratic model to the data.(c) Based on your model, what was the initial position of the shuttle?
(d) Using your model, estimate the velocity of the shuttle when t = 0.1 and when t = 0.45 seconds.(e) What is your estimated acceleration of the shuttle when t = 0.1 and when t = 0.45 seconds?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 9

The linear approximation of a function is useful only if the change in x is small. Illustrate this fact by approximating $\sqrt { 18 }$ by regarding 18 to be "near" 36 instead of 16.

Accepted Answer

@#IMG-DLM& @#IMG-DLM& and @#IMG-DLM& @#IMG-DLM& , so @#IMG-DLM& , and @#IMG-DLM& . From a

Question 10

Find the x-coordinate of the point at which the graph of $y = x \ln x$ has a horizontal tangent.

Accepted Answer

The answer of Find the x-coordinate of the point at...

Question 11

Find $\frac { d y } { d x }$ if $x ^ { 2 } - 2 x y ^ { 2 } = y ^ { 4 }$ .

Accepted Answer

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

Question 12

Given $f ( x ) = \left( 4 - x ^ { 2 } \right) e ^ { x }$ find all values of x where $f ^ { \prime } ( x ) = 0$

Accepted Answer

The answer of Given \(f ( x ) = \left(...

Question 13

A particle moves along a straight line with equation of motion $S = t ^ { 3 } = 2 t$ . Find the smallest value of its velocity (for t ≥ 0).

Accepted Answer

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

Question 14

Find the slope of the tangent to the curve $x = t ^ { 3 }$ , $y = t ^ { 4 }$ when $t = 3$

Accepted Answer

A)  $\frac { 9 } { 4 }$ 
B)  $\frac { 3 } { 16 }$ 
C)  $\frac { 4 } { 9 }$ 
D)  $\frac { 4 } { 3 }$ 
E) 3 
F) $\frac { 3 } { 4 }$
G) $\frac { 16 } { 3 }$
H)4 
A)  $\frac { 9 } { 4 }$ 
B)  $\frac { 3 } { 16 }$ 
C)  $\frac { 4 } { 9 }$ 
D)  $\frac { 4 } { 3 }$ 
E) 3 
F) $\frac { 3 } { 4 }$
G) $\frac { 16 } { 3 }$
H)4

Question 15

Given $f ( 4 ) = 2$ , $f ^ { \prime } ( 4 ) = - 1.5$ , $g ( 4 ) = - 3$ and $g ^ { \prime } ( 4 ) = 2.3$ , find the value of $\left( \frac { 9 f } { 2 g } \right) ^ { \prime }(4)$ .

Accepted Answer

A)  $- 4.5$ 
B)  $- 2.3$ 
C)  $- 0.1$ 
D)  $- 20$ 
E)  $- 5$
F) $- 0.22$

G) $- 0.45$

H) $- 0.05$ 
A)  $- 4.5$ 
B)  $- 2.3$ 
C)  $- 0.1$ 
D)  $- 20$ 
E)  $- 5$
F) $- 0.22$

G) $- 0.45$

H) $- 0.05$

Question 16

Find the derivative of $f ( x ) = \frac { x } { x ^ { 2 } + 3 }$ .

Accepted Answer

A)  $\frac { 1 } { 2 x + 3 }$ 
B)  $\frac { - 3 + x ^ { 2 } } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$ 
C)  $\frac { 3 - x ^ { 2 } } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$ 
D)  $\frac { 3 - x ^ { 2 } } { \left( x ^ { 2 } + 3 \right) }$ 
E)  $\frac { 3 } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$
F) $\frac { 1 } { 2 x }$
 
G) $\frac { 3 - x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$

H)None of these 
A)  $\frac { 1 } { 2 x + 3 }$ 
B)  $\frac { - 3 + x ^ { 2 } } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$ 
C)  $\frac { 3 - x ^ { 2 } } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$ 
D)  $\frac { 3 - x ^ { 2 } } { \left( x ^ { 2 } + 3 \right) }$ 
E)  $\frac { 3 } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$
F) $\frac { 1 } { 2 x }$
 
G) $\frac { 3 - x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$

H)None of these

Question 17

If $f ( x ) = \sin \pi x , \text { find the value of } f ^ { ( 23 ) } ( x )$

Accepted Answer

A)  $\pi ^ { 23 } \sin x$ 
B)  $\pi ^ { 23 } \cos x$ 
C)  $- \pi ^ { 23 } \sin x$ 
D)  $- \pi ^ { 23 } \cos x$ 
E)  $\pi ^ { 23 } \sin \pi x$
F) $\pi ^ { 23 } \cos \pi x$
G) $- \pi ^ { 23 } \sin \pi x$
H) $- \pi ^ { 23 } \cos \pi x$ 
A)  $\pi ^ { 23 } \sin x$ 
B)  $\pi ^ { 23 } \cos x$ 
C)  $- \pi ^ { 23 } \sin x$ 
D)  $- \pi ^ { 23 } \cos x$ 
E)  $\pi ^ { 23 } \sin \pi x$
F) $\pi ^ { 23 } \cos \pi x$
G) $- \pi ^ { 23 } \sin \pi x$
H) $- \pi ^ { 23 } \cos \pi x$

Question 18

The following table shows the concentration (in mol/L) of a certain chemical in terms of reaction time (in hours) during a decomposition reaction. $$\begin{array} { | l | c | c | c | c | c | c | } 
\hline \text { Time (hours) } & 0 & 5 & 10 & 20 & 30 & 50 \
\hline \text { Concentration (mol/L) } & 2.32 & 1.86 & 1.49 & 0.98 & 0.62 & 0.25 \
\hline
\end{array}$$ (a) Find the average rate of change of concentration with respect to time for the following time intervals:
(i) [0, 5]
(ii) [10, 20]
(iii) [30, 50]
(b) Plot the points from the table and fit an appropriate exponential model to the data.(c) From your model in part (b), determine the instantaneous rate of change of concentration with respect to time.(d) Is the rate of change of concentration increasing or decreasing with respect to time? Justify your answer.

Accepted Answer

(a) ( i) 0.092 mol/L/h
(ii) -0

Question 19

Suppose $f ( x ) = ( \sec x + \tan x ) ( \sec x - \tan x )$ (a) Use the product rule to differentiate $f ( x )$ (b) What does the value of the derivative tell you?
(c) How can you arrive at this simple result more quickly?

Accepted Answer

(a) @#IMG-DLM& (b) Since @#IMG-DLM& , f(x) must be

Question 20

Find the derivative of $f ( x ) = x ^ { 2 } \cos x$ .

Accepted Answer

A) 2x cos x 
B)  $2 x \cos x - x ^ { 2 } \sin x$ 
C)  $- 2 x \sin x$ 
D)  $2 x \cos x + x ^ { 2 } \sin x$ 
E)  $2 x \sin x$ 
F) $3 x ^ { 2 } \cos \left( x ^ { 3 } \right)$
G) $- x ^ { 2 } \sin x$
 H)None of the above 
A) 2x cos x 
B)  $2 x \cos x - x ^ { 2 } \sin x$ 
C)  $- 2 x \sin x$ 
D)  $2 x \cos x + x ^ { 2 } \sin x$ 
E)  $2 x \sin x$ 
F) $3 x ^ { 2 } \cos \left( x ^ { 3 } \right)$
G) $- x ^ { 2 } \sin x$
 H)None of the above

Given $f ( x ) = e ^ { x } - x ^ { 3 }$ , find $f ^ { \prime } ( x ) , f ^ { \prime \prime } ( x )$ and $f ^ { ( 3 ) } ( x )$ .

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( x ^ { 2 } f ( x ) \right) \text { when } x = 4 \text {. }$

Find the x-coordinate of the point at which the graph of $y = \frac { \ln x } { x }$ has a horizontal tangent.

Find the equation of the line normal to the curve defined by the equation $x ^ { 3 } y ^ { 4 } - 5 = x ^ { 3 } - x ^ { 2 } + y$ at the point (2, $- 1$ ).

Find the point(s) where the curve $2 x ^ { 2 } - 2 x y + y ^ { 2 } = 4$ has a horizontal tangent.

Differentiate each of the following functions: (a) $f ( x ) = x ^ { 3 } \cos x$ (b) $f ( x ) = \frac { x } { \sin x }$ (c) $f ( x ) = \frac { x \sin x } { x + \sin x }$

The linear approximation of a function is useful only if the change in x is small. Illustrate this fact by approximating $\sqrt { 18 }$ by regarding 18 to be "near" 36 instead of 16.

Find the x-coordinate of the point at which the graph of $y = x \ln x$ has a horizontal tangent.

Find $\frac { d y } { d x }$ if $x ^ { 2 } - 2 x y ^ { 2 } = y ^ { 4 }$ .

Given $f ( x ) = \left( 4 - x ^ { 2 } \right) e ^ { x }$ find all values of x where $f ^ { \prime } ( x ) = 0$

A particle moves along a straight line with equation of motion $S = t ^ { 3 } = 2 t$ . Find the smallest value of its velocity (for t ≥ 0).

Find the slope of the tangent to the curve $x = t ^ { 3 }$ , $y = t ^ { 4 }$ when $t = 3$

Given $f ( 4 ) = 2$ , $f ^ { \prime } ( 4 ) = - 1.5$ , $g ( 4 ) = - 3$ and $g ^ { \prime } ( 4 ) = 2.3$ , find the value of $\left( \frac { 9 f } { 2 g } \right) ^ { \prime }(4)$ .

Find the derivative of $f ( x ) = \frac { x } { x ^ { 2 } + 3 }$ .

If $f ( x ) = \sin \pi x , \text { find the value of } f ^ { ( 23 ) } ( x )$

Suppose $f ( x ) = ( \sec x + \tan x ) ( \sec x - \tan x )$ (a) Use the product rule to differentiate $f ( x )$ (b) What does the value of the derivative tell you? (c) How can you arrive at this simple result more quickly?

Find the derivative of $f ( x ) = x ^ { 2 } \cos x$ .

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Exam 3: Differentiation Rules

Given f(x)=ex−x3f ( x ) = e ^ { x } - x ^ { 3 }f(x)=ex−x3 , find f′(x),f′′(x)f ^ { \prime } ( x ) , f ^ { \prime \prime } ( x )f′(x),f′′(x) and f(3)(x)f ^ { ( 3 ) } ( x )f(3)(x) .

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find ddx(x2f(x)) when x=4. \frac { d } { d x } \left( x ^ { 2 } f ( x ) \right) \text { when } x = 4 \text {. }dxd​(x2f(x)) when x=4.

Find the x-coordinate of the point at which the graph of y=ln⁡xxy = \frac { \ln x } { x }y=xlnx​ has a horizontal tangent.

Find the equation of the line normal to the curve defined by the equation x3y4−5=x3−x2+yx ^ { 3 } y ^ { 4 } - 5 = x ^ { 3 } - x ^ { 2 } + yx3y4−5=x3−x2+y at the point (2, −1- 1−1 ).

Find the point(s) where the curve 2x2−2xy+y2=42 x ^ { 2 } - 2 x y + y ^ { 2 } = 42x2−2xy+y2=4 has a horizontal tangent.

Differentiate each of the following functions: (a) f(x)=x3cos⁡xf ( x ) = x ^ { 3 } \cos xf(x)=x3cosx (b) f(x)=xsin⁡xf ( x ) = \frac { x } { \sin x }f(x)=sinxx​ (c) f(x)=xsin⁡xx+sin⁡xf ( x ) = \frac { x \sin x } { x + \sin x }f(x)=x+sinxxsinx​

The linear approximation of a function is useful only if the change in x is small. Illustrate this fact by approximating 18\sqrt { 18 }18​ by regarding 18 to be "near" 36 instead of 16.

Find the x-coordinate of the point at which the graph of y=xln⁡xy = x \ln xy=xlnx has a horizontal tangent.

Find dydx\frac { d y } { d x }dxdy​ if x2−2xy2=y4x ^ { 2 } - 2 x y ^ { 2 } = y ^ { 4 }x2−2xy2=y4 .

Given f(x)=(4−x2)exf ( x ) = \left( 4 - x ^ { 2 } \right) e ^ { x }f(x)=(4−x2)ex find all values of x where f′(x)=0f ^ { \prime } ( x ) = 0f′(x)=0

A particle moves along a straight line with equation of motion S=t3=2tS = t ^ { 3 } = 2 tS=t3=2t . Find the smallest value of its velocity (for t ≥ 0).

Find the slope of the tangent to the curve x=t3x = t ^ { 3 }x=t3 , y=t4y = t ^ { 4 }y=t4 when t=3t = 3t=3

Given f(4)=2f ( 4 ) = 2f(4)=2 , f′(4)=−1.5f ^ { \prime } ( 4 ) = - 1.5f′(4)=−1.5 , g(4)=−3g ( 4 ) = - 3g(4)=−3 and g′(4)=2.3g ^ { \prime } ( 4 ) = 2.3g′(4)=2.3 , find the value of (9f2g)′(4)\left( \frac { 9 f } { 2 g } \right) ^ { \prime }(4)(2g9f​)′(4) .

Find the derivative of f(x)=xx2+3f ( x ) = \frac { x } { x ^ { 2 } + 3 }f(x)=x2+3x​ .

If f(x)=sin⁡πx, find the value of f(23)(x)f ( x ) = \sin \pi x , \text { find the value of } f ^ { ( 23 ) } ( x )f(x)=sinπx, find the value of f(23)(x)

Suppose f(x)=(sec⁡x+tan⁡x)(sec⁡x−tan⁡x)f ( x ) = ( \sec x + \tan x ) ( \sec x - \tan x )f(x)=(secx+tanx)(secx−tanx) (a) Use the product rule to differentiate f(x)f ( x )f(x) (b) What does the value of the derivative tell you? (c) How can you arrive at this simple result more quickly?

Find the derivative of f(x)=x2cos⁡xf ( x ) = x ^ { 2 } \cos xf(x)=x2cosx .

Functions and Models

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Given $f ( x ) = e ^ { x } - x ^ { 3 }$ , find $f ^ { \prime } ( x ) , f ^ { \prime \prime } ( x )$ and $f ^ { ( 3 ) } ( x )$ .

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( x ^ { 2 } f ( x ) \right) \text { when } x = 4 \text {. }$

Find the x-coordinate of the point at which the graph of $y = \frac { \ln x } { x }$ has a horizontal tangent.

Find the equation of the line normal to the curve defined by the equation $x ^ { 3 } y ^ { 4 } - 5 = x ^ { 3 } - x ^ { 2 } + y$ at the point (2, $- 1$ ).

Find the point(s) where the curve $2 x ^ { 2 } - 2 x y + y ^ { 2 } = 4$ has a horizontal tangent.

Differentiate each of the following functions: (a) $f ( x ) = x ^ { 3 } \cos x$ (b) $f ( x ) = \frac { x } { \sin x }$ (c) $f ( x ) = \frac { x \sin x } { x + \sin x }$

The linear approximation of a function is useful only if the change in x is small. Illustrate this fact by approximating $\sqrt { 18 }$ by regarding 18 to be "near" 36 instead of 16.

Find the x-coordinate of the point at which the graph of $y = x \ln x$ has a horizontal tangent.

Find $\frac { d y } { d x }$ if $x ^ { 2 } - 2 x y ^ { 2 } = y ^ { 4 }$ .

Given $f ( x ) = \left( 4 - x ^ { 2 } \right) e ^ { x }$ find all values of x where $f ^ { \prime } ( x ) = 0$

A particle moves along a straight line with equation of motion $S = t ^ { 3 } = 2 t$ . Find the smallest value of its velocity (for t ≥ 0).

Find the slope of the tangent to the curve $x = t ^ { 3 }$ , $y = t ^ { 4 }$ when $t = 3$

Given $f ( 4 ) = 2$ , $f ^ { \prime } ( 4 ) = - 1.5$ , $g ( 4 ) = - 3$ and $g ^ { \prime } ( 4 ) = 2.3$ , find the value of $\left( \frac { 9 f } { 2 g } \right) ^ { \prime }(4)$ .

Find the derivative of $f ( x ) = \frac { x } { x ^ { 2 } + 3 }$ .

If $f ( x ) = \sin \pi x , \text { find the value of } f ^ { ( 23 ) } ( x )$

Suppose $f ( x ) = ( \sec x + \tan x ) ( \sec x - \tan x )$ (a) Use the product rule to differentiate $f ( x )$ (b) What does the value of the derivative tell you? (c) How can you arrive at this simple result more quickly?

Find the derivative of $f ( x ) = x ^ { 2 } \cos x$ .