Question 1

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

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 
$e ^ { \frac { \pi } { 2 } }$ 
E)  $e ^ { x }$ 
F) $\frac { \pi } { 2 }$ 
G) $\pi$
H)2 $\pi$ 
A) 0 
B) 1 
C) 2 
D) 
$e ^ { \frac { \pi } { 2 } }$ 
E)  $e ^ { x }$ 
F) $\frac { \pi } { 2 }$ 
G) $\pi$
H)2 $\pi$

Question 2

Find the slope of the tangent line to the curve y = $\frac { 1 } { 1 + 2 x }$ at the point (1, $\frac { 1 } { 3 }$ ).

Accepted Answer

A)  $- \frac { 1 } { 4 }$ 
B) 2 
C) 
$- 1$ 
D)  $\frac { 1 } { 4 }$ 
E)  $- \frac { 2 } { 9 }$

F) $- 2$

G)1
H) $\frac { 1 } { 2 }$ 
A)  $- \frac { 1 } { 4 }$ 
B) 2 
C) 
$- 1$ 
D)  $\frac { 1 } { 4 }$ 
E)  $- \frac { 2 } { 9 }$

F) $- 2$

G)1
H) $\frac { 1 } { 2 }$

Question 3

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = x + 2 \cos x , 0 \leq x \leq 2 \pi$ has zero slope.

Accepted Answer

The answer of Find the x-coordinate(s) of the point(s) where...

Question 4

Differentiate the following functions:
(a) $f ( x ) = ( \ln x ) ^ { 3 }$ (b) $f ( x ) = \sqrt { x } \ln x$ (c) $f ( x ) = \ln \left( \frac { \tan x } { x ^ { 2 } + 1 } \right)$

Accepted Answer

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

Question 5

Let $f ( x ) = ( \sqrt { x } ) ^ { x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

Accepted Answer

A) 0 
B)  $\frac { 1 } { 4 }$ f.2 
C)  $\frac { 1 } { 2 }$ g.4 
D)  $\frac { 3 } { 4 }$ h.6 
E) 1 
A) 0 
B)  $\frac { 1 } { 4 }$ f.2 
C)  $\frac { 1 } { 2 }$ g.4 
D)  $\frac { 3 } { 4 }$ h.6 
E) 1

Question 6

Let $f ( x ) = x ^ { 3 } - 4 x ^ { 2 }$ (a) Find a linear approximation of f at x = 1.(b) Use this linear approximation to predict the value of the function at -1, 0, 0.9, 1.1, 2, and 3.(c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval [-1; 3]. What does the graph tell you about the size of the difference between the function values and the linear approximation values?

Accepted Answer

(a) @#IMG-DLM& (b), (c) @#IMG-DLM& The linear appr

Question 7

Find the derivative of $f ( x ) = \frac { 1 } { ( 1 - 2 x ) ^ { 3 } }$ .

Accepted Answer

A)  $\frac { 6 } { ( 1 - 2 x ) ^ { 4 } }$ 
B)  $- \frac { 6 } { ( 1 - 2 x ) ^ { 4 } }$ 
C)  $\frac { 3 } { ( 1 - 2 x ) ^ { 4 } }$ 
D)  $- \frac { 1 } { 6 ( 1 - 2 x ) ^ { 2 } }$ 
E)  $- \frac { 3 } { ( 1 - 2 x ) ^ { 4 } }$
F) $\frac { 1 } { 3 ( 1 - 2 x ) ^ { 2 } }$
G) $- \frac { 2 } { ( 1 - 2 x ) ^ { 4 } }$
H)None of these 
A)  $\frac { 6 } { ( 1 - 2 x ) ^ { 4 } }$ 
B)  $- \frac { 6 } { ( 1 - 2 x ) ^ { 4 } }$ 
C)  $\frac { 3 } { ( 1 - 2 x ) ^ { 4 } }$ 
D)  $- \frac { 1 } { 6 ( 1 - 2 x ) ^ { 2 } }$ 
E)  $- \frac { 3 } { ( 1 - 2 x ) ^ { 4 } }$
F) $\frac { 1 } { 3 ( 1 - 2 x ) ^ { 2 } }$
G) $- \frac { 2 } { ( 1 - 2 x ) ^ { 4 } }$
H)None of these

Question 8

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 25.05 }$ .

Accepted Answer

A) 5.001 
B) 5.002 
C) 5.003 
D) 5.004 
E) 5.005
F)5.006
G)5.007
H)5.008 
A) 5.001 
B) 5.002 
C) 5.003 
D) 5.004 
E) 5.005
F)5.006
G)5.007
H)5.008

Question 9

If $\sqrt { x + y } + \sqrt { x - y } = 4$ find the value of $\frac { d y } { d x }$ at the point (5, 4).

Accepted Answer

The answer of If \(\sqrt { x + y }...

Question 10

Simplify the express: $\sec \left( \tan ^ { - 1 } x \right)$ .

Accepted Answer

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

Question 11

Let $y = x ^ { 2 }$ , x = 3, and $\Delta$x = 1. Find the value of the corresponding change $\Delta$y.

Accepted Answer

A) 4 
B) 8 
C) 7 
D) 2 
E) 3
F)6
G)1
H)5 
A) 4 
B) 8 
C) 7 
D) 2 
E) 3
F)6
G)1
H)5

Question 12

Find an equation of the tangent line to $y = x ^ { 4 } - x ^ { 3 }$ at (-1, 2).

Accepted Answer

@#IMG-DLM& . At (-1, 2), @#IMG-DLM& . Th

Question 13

Let f (x) = $\cos ^ { 2 } x$ .(a) Find a linear approximation of f at x = $\frac { \pi } { 3 } .$ (b) Use this linear approximation to predict the value of the function at $\frac { \pi } { 3 } - 1 , \frac { \pi } { 3 } - 0.1 , \frac { \pi } { 3 } + 1 \text {, }$ and $\frac { \pi } { 3 } + 0.1$ (c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval $\left[ \frac { \pi } { 3 } - 1 , \frac { \pi } { 3 } + 1 \right]$ What does the graph tell you about the size of the difference between the function values and the linear approximation values?

Accepted Answer

(a) @#IMG-DLM& (b), (c) @#IMG-DLM& The l

Question 14

Let $f ( x ) = \log _ { 3 } x ^ { 2 }$ . Find the value of $f ^ { \prime } ( 2 )$ .

Accepted Answer

A)  $3 ^ { e }$ 
B) ln 3 
C) ln $\left( \frac { 1 } { 3 } \right)$ 
D) 1 
E)  $\frac { 1 } { \sqrt { 3 } }$
F) $\frac { 1 } { \ln 3 }$
G)
$\frac { 1 } { 3 }$ 
H) $\sqrt { 3 }$ 
A)  $3 ^ { e }$ 
B) ln 3 
C) ln $\left( \frac { 1 } { 3 } \right)$ 
D) 1 
E)  $\frac { 1 } { \sqrt { 3 } }$
F) $\frac { 1 } { \ln 3 }$
G)
$\frac { 1 } { 3 }$ 
H) $\sqrt { 3 }$

Question 15

Let $f ( x ) = e ^ { - 1 / x }$ Find the value of $f ^ { \prime } ( 1 )$

Accepted Answer

A) 2e 
B) e 
C)  $2 e ^ { - 1 }$ 
D)  $- 2 e$ 
E)  $e ^ { - 1 }$ 
F) $- e$
G) $- 2 e ^ { - 1 }$ 
H) $- e ^ { - 1 }$ 
A) 2e 
B) e 
C)  $2 e ^ { - 1 }$ 
D)  $- 2 e$ 
E)  $e ^ { - 1 }$ 
F) $- e$
G) $- 2 e ^ { - 1 }$ 
H) $- e ^ { - 1 }$

Question 16

Given $f ( 3 ) = 5$ , $f ^ { \prime } ( 3 ) = 1.1$ , $g ( 3 ) = - 4$ and $g ^ { \prime } ( 3 ) = 0.7$ , find the value of $( f - 2 g ) ^ { \prime } ( 3 )$ .

Accepted Answer

A) 1.8 
B) 0.3 
C)  $- 0.9$ 
D)  $- 1.8$ 
E) 0.9
F) $- 0.3$
G)3.5
H) $- 3.5$ 
A) 1.8 
B) 0.3 
C)  $- 0.9$ 
D)  $- 1.8$ 
E) 0.9
F) $- 0.3$
G)3.5
H) $- 3.5$

Question 17

A particle moves along a straight line with equation of motion $s = t ^ { 3 } - t ^ { 2 }$ . Find the value of t at which the acceleration is equal to zero.

Accepted Answer

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

Question 18

The cost function of manufacturing x meters of a fabric is $C ( x ) = 20,000 + 5 x - 0.004 x ^ { 2 } + 0.000003 x ^ { 3 }$ .

Accepted Answer

(a) Find the margina

Question 19

Let $f ( x ) = x ^ { 2 x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

Accepted Answer

A) 2 
B) e - 1 
C)  $e ^ { e } - 1$ 
D) e + 1 
E) e
F) $e ^ { e + 1 }$ 
G) $e ^ { e - 1 }$ 
H) $e ^ { 2 }$ 
A) 2 
B) e - 1 
C)  $e ^ { e } - 1$ 
D) e + 1 
E) e
F) $e ^ { e + 1 }$ 
G) $e ^ { e - 1 }$ 
H) $e ^ { 2 }$

Question 20

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

Accepted Answer

A) 2cos x 
B) cos( $x ^ { 2 }$ ) 
C) 2sin x 
D)  $\cos ^ { 2 } x$ 
E)  $2 \cos ^ { 2 } x$ 
F) $2 \cos \left( x ^ { 2 } \right)$
G) $2 \sin x \cos x$
H) $- 2 \sin x \cos x$ 
A) 2cos x 
B) cos( $x ^ { 2 }$ ) 
C) 2sin x 
D)  $\cos ^ { 2 } x$ 
E)  $2 \cos ^ { 2 } x$ 
F) $2 \cos \left( x ^ { 2 } \right)$
G) $2 \sin x \cos x$
H) $- 2 \sin x \cos x$

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

Find the slope of the tangent line to the curve y = $\frac { 1 } { 1 + 2 x }$ at the point (1, $\frac { 1 } { 3 }$ ).

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = x + 2 \cos x , 0 \leq x \leq 2 \pi$ has zero slope.

Differentiate the following functions: (a) $f ( x ) = ( \ln x ) ^ { 3 }$ (b) $f ( x ) = \sqrt { x } \ln x$ (c) $f ( x ) = \ln \left( \frac { \tan x } { x ^ { 2 } + 1 } \right)$

Let $f ( x ) = ( \sqrt { x } ) ^ { x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

Find the derivative of $f ( x ) = \frac { 1 } { ( 1 - 2 x ) ^ { 3 } }$ .

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 25.05 }$ .

If $\sqrt { x + y } + \sqrt { x - y } = 4$ find the value of $\frac { d y } { d x }$ at the point (5, 4).

Simplify the express: $\sec \left( \tan ^ { - 1 } x \right)$ .

Let $y = x ^ { 2 }$ , x = 3, and $\Delta$ x = 1. Find the value of the corresponding change $\Delta$ y.

Find an equation of the tangent line to $y = x ^ { 4 } - x ^ { 3 }$ at (-1, 2).

Let $f ( x ) = \log _ { 3 } x ^ { 2 }$ . Find the value of $f ^ { \prime } ( 2 )$ .

Let $f ( x ) = e ^ { - 1 / x }$ Find the value of $f ^ { \prime } ( 1 )$

Given $f ( 3 ) = 5$ , $f ^ { \prime } ( 3 ) = 1.1$ , $g ( 3 ) = - 4$ and $g ^ { \prime } ( 3 ) = 0.7$ , find the value of $( f - 2 g ) ^ { \prime } ( 3 )$ .

A particle moves along a straight line with equation of motion $s = t ^ { 3 } - t ^ { 2 }$ . Find the value of t at which the acceleration is equal to zero.

The cost function of manufacturing x meters of a fabric is $C ( x ) = 20,000 + 5 x - 0.004 x ^ { 2 } + 0.000003 x ^ { 3 }$ .

Let $f ( x ) = x ^ { 2 x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

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

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Exam 3: Differentiation Rules

Let f(x)=(sin⁡x)xf ( x ) = ( \sin x ) ^ { x }f(x)=(sinx)x . Find the value of f′(π2)f ^ { \prime } \left( \frac { \pi } { 2 } \right)f′(2π​) .

Find the slope of the tangent line to the curve y = 11+2x\frac { 1 } { 1 + 2 x }1+2x1​ at the point (1, 13\frac { 1 } { 3 }31​ ).

Find the x-coordinate(s) of the point(s) where the tangent to the curve y=x+2cos⁡x,0≤x≤2πy = x + 2 \cos x , 0 \leq x \leq 2 \piy=x+2cosx,0≤x≤2π has zero slope.

Differentiate the following functions: (a) f(x)=(ln⁡x)3f ( x ) = ( \ln x ) ^ { 3 }f(x)=(lnx)3 (b) f(x)=xln⁡xf ( x ) = \sqrt { x } \ln xf(x)=x​lnx (c) f(x)=ln⁡(tan⁡xx2+1)f ( x ) = \ln \left( \frac { \tan x } { x ^ { 2 } + 1 } \right)f(x)=ln(x2+1tanx​)

Let f(x)=(x)xf ( x ) = ( \sqrt { x } ) ^ { x }f(x)=(x​)x . Find the value of f′(1)f ^ { \prime } ( 1 )f′(1) .

Find the derivative of f(x)=1(1−2x)3f ( x ) = \frac { 1 } { ( 1 - 2 x ) ^ { 3 } }f(x)=(1−2x)31​ .

Find the linear approximation of the function f (x) = x+24\sqrt { x + 24 }x+24​ at x 111 = 1 and use it to approximate 25.05\sqrt { 25.05 }25.05​ .

If x+y+x−y=4\sqrt { x + y } + \sqrt { x - y } = 4x+y​+x−y​=4 find the value of dydx\frac { d y } { d x }dxdy​ at the point (5, 4).

Simplify the express: sec⁡(tan⁡−1x)\sec \left( \tan ^ { - 1 } x \right)sec(tan−1x) .

Let y=x2y = x ^ { 2 }y=x2 , x = 3, and Δ\DeltaΔ x = 1. Find the value of the corresponding change Δ\DeltaΔ y.

Find an equation of the tangent line to y=x4−x3y = x ^ { 4 } - x ^ { 3 }y=x4−x3 at (-1, 2).

Let f(x)=log⁡3x2f ( x ) = \log _ { 3 } x ^ { 2 }f(x)=log3​x2 . Find the value of f′(2)f ^ { \prime } ( 2 )f′(2) .

Let f(x)=e−1/xf ( x ) = e ^ { - 1 / x }f(x)=e−1/x Find the value of f′(1)f ^ { \prime } ( 1 )f′(1)

Given f(3)=5f ( 3 ) = 5f(3)=5 , f′(3)=1.1f ^ { \prime } ( 3 ) = 1.1f′(3)=1.1 , g(3)=−4g ( 3 ) = - 4g(3)=−4 and g′(3)=0.7g ^ { \prime } ( 3 ) = 0.7g′(3)=0.7 , find the value of (f−2g)′(3)( f - 2 g ) ^ { \prime } ( 3 )(f−2g)′(3) .

A particle moves along a straight line with equation of motion s=t3−t2s = t ^ { 3 } - t ^ { 2 }s=t3−t2 . Find the value of t at which the acceleration is equal to zero.

The cost function of manufacturing x meters of a fabric is C(x)=20,000+5x−0.004x2+0.000003x3C ( x ) = 20,000 + 5 x - 0.004 x ^ { 2 } + 0.000003 x ^ { 3 }C(x)=20,000+5x−0.004x2+0.000003x3 .

Let f(x)=x2xf ( x ) = x ^ { 2 x }f(x)=x2x . Find the value of f′(1)f ^ { \prime } ( 1 )f′(1) .

Find the derivative of f(x)=sin⁡2xf ( x ) = \sin ^ { 2 } xf(x)=sin2x .

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

Find the slope of the tangent line to the curve y = $\frac { 1 } { 1 + 2 x }$ at the point (1, $\frac { 1 } { 3 }$ ).

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = x + 2 \cos x , 0 \leq x \leq 2 \pi$ has zero slope.

Differentiate the following functions: (a) $f ( x ) = ( \ln x ) ^ { 3 }$ (b) $f ( x ) = \sqrt { x } \ln x$ (c) $f ( x ) = \ln \left( \frac { \tan x } { x ^ { 2 } + 1 } \right)$

Let $f ( x ) = ( \sqrt { x } ) ^ { x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

Find the derivative of $f ( x ) = \frac { 1 } { ( 1 - 2 x ) ^ { 3 } }$ .

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 25.05 }$ .

If $\sqrt { x + y } + \sqrt { x - y } = 4$ find the value of $\frac { d y } { d x }$ at the point (5, 4).

Simplify the express: $\sec \left( \tan ^ { - 1 } x \right)$ .

Let $y = x ^ { 2 }$ , x = 3, and $\Delta$ x = 1. Find the value of the corresponding change $\Delta$ y.

Find an equation of the tangent line to $y = x ^ { 4 } - x ^ { 3 }$ at (-1, 2).

Let $f ( x ) = \log _ { 3 } x ^ { 2 }$ . Find the value of $f ^ { \prime } ( 2 )$ .

Let $f ( x ) = e ^ { - 1 / x }$ Find the value of $f ^ { \prime } ( 1 )$

Given $f ( 3 ) = 5$ , $f ^ { \prime } ( 3 ) = 1.1$ , $g ( 3 ) = - 4$ and $g ^ { \prime } ( 3 ) = 0.7$ , find the value of $( f - 2 g ) ^ { \prime } ( 3 )$ .

A particle moves along a straight line with equation of motion $s = t ^ { 3 } - t ^ { 2 }$ . Find the value of t at which the acceleration is equal to zero.

The cost function of manufacturing x meters of a fabric is $C ( x ) = 20,000 + 5 x - 0.004 x ^ { 2 } + 0.000003 x ^ { 3 }$ .

Let $f ( x ) = x ^ { 2 x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

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