Question 1

Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \tan 2 x } { x }$ .

Accepted Answer

A) 4 
B) 1 
C)  $\frac { 1 } { 4 }$ 
D) 2 
E)  $\frac { 1 } { 2 }$ 
F)0
G) $\frac { 1 } { 3 }$ 
H)3 
A) 4 
B) 1 
C)  $\frac { 1 } { 4 }$ 
D) 2 
E)  $\frac { 1 } { 2 }$ 
F)0
G) $\frac { 1 } { 3 }$ 
H)3 D

Question 2

Differentiate the following functions:
(a) $f ( x ) = \frac { e ^ { x } } { x }$ (b) $f ( x ) = ( \sqrt { x } + 2 ) ( 2 \sqrt { x } - 4 )$ (c) $f ( x ) = e + 3 x ^ { e }$ (d ) $f ( x ) = \pi ^ { e } \left( x ^ { 2 } - x \right)$

Accepted Answer

(a) $f ^ { \prime } ( x ) = \frac { x e ^ { x } - e ^ { x } } { x ^ { 2 } }$ (b) $f ^ { \prime } ( x ) = 2$ (c) $f ^ { \prime } ( x ) = 3 e x ^ { e - 1 }$ (d) $f ^ { \prime } ( x ) = \pi ^ { e } ( 2 x - 1 )$

Question 3

Use implicit differentiation to find $y ^ { \prime \prime } \text { if } 2 x y = y ^ { 2 }$ .

Accepted Answer

$y ^ { \prime \prime } = \frac { y ^ { 2 } - 2 x y } { ( y - x ) ^ { 3 } }$

Question 4

Find the derivative of $f ( t ) = \sqrt { t ^ { 5 } } + \sqrt [ 5 ] { t ^ { 2 } }$ .

Accepted Answer

A)  $5 t ^ { - 1 / 2 } + 2 t ^ { - 3 / 2 }$ 
B)  $\frac { 5 } { 2 } t ^ { 1 / 2 } - \frac { 2 } { 5 } t ^ { 3 / 5 }$ 
C)  $\frac { 5 } { 2 } t ^ { 3 / 2 } + \frac { 2 } { 5 } t ^ { - 3 / 5 }$ 
D)  $\frac { 2 } { 5 } t ^ { 3 / 2 } + \frac { 5 } { 2 } t ^ { - 3 / 5 }$ 
E)  $5 \sqrt { t ^ { 4 } } + 2 \sqrt [ 5 ] { t }$ 
F) $\frac { 5 } { 2 } t ^ { 4 } - \frac { 2 } { 5 } t$

G) $\frac { 3 } { 2 } t ^ { 5 / 2 } - \frac { 2 } { 3 } t ^ { 3 / 2 }$

H)None of the above 
A)  $5 t ^ { - 1 / 2 } + 2 t ^ { - 3 / 2 }$ 
B)  $\frac { 5 } { 2 } t ^ { 1 / 2 } - \frac { 2 } { 5 } t ^ { 3 / 5 }$ 
C)  $\frac { 5 } { 2 } t ^ { 3 / 2 } + \frac { 2 } { 5 } t ^ { - 3 / 5 }$ 
D)  $\frac { 2 } { 5 } t ^ { 3 / 2 } + \frac { 5 } { 2 } t ^ { - 3 / 5 }$ 
E)  $5 \sqrt { t ^ { 4 } } + 2 \sqrt [ 5 ] { t }$ 
F) $\frac { 5 } { 2 } t ^ { 4 } - \frac { 2 } { 5 } t$

G) $\frac { 3 } { 2 } t ^ { 5 / 2 } - \frac { 2 } { 3 } t ^ { 3 / 2 }$

H)None of the above

Question 5

Suppose that $h ( x ) = f ( g ( x ) )$ and $g ( 3 ) = 6$ , $g ^ { \prime } ( 3 ) = 4 ,$ $f ^ { \prime } ( 3 ) = 2 , f ^ { \prime } ( 6 ) = 7$ Find the value of $h ^ { \prime } ( 3 ) \text {. }$

Accepted Answer

A) 4 
B) 8 
C) 12 
D) 16 
E) 20
F)24
G)28
H)32 
A) 4 
B) 8 
C) 12 
D) 16 
E) 20
F)24
G)28
H)32

Question 6

Let $f ( x ) = \sin ^ { - 1 } \left( \frac { 4 } { x } \right)$ . Find the value of $f ^ { \prime } ( 8 )$ .

Accepted Answer

A)  $- \frac { 1 } { 8 \sqrt { 3 } }$ 
B)  $- \frac { 1 } { 4 \sqrt { 3 } }$ 
C)  $- \frac { 1 } { 2 \sqrt { 3 } }$ 
D)  $- \frac { 1 } { \sqrt { 3 } }$ 
E)  $\frac { \pi } { 8 }$
F) $\frac { \pi } { 4 }$ 
G) $\frac { \pi } { 2 }$
H) $\pi$ 
A)  $- \frac { 1 } { 8 \sqrt { 3 } }$ 
B)  $- \frac { 1 } { 4 \sqrt { 3 } }$ 
C)  $- \frac { 1 } { 2 \sqrt { 3 } }$ 
D)  $- \frac { 1 } { \sqrt { 3 } }$ 
E)  $\frac { \pi } { 8 }$
F) $\frac { \pi } { 4 }$ 
G) $\frac { \pi } { 2 }$
H) $\pi$

Question 7

At what value of t does the curve $x = 2 t - 3 t ^ { 2 } , y = t ^ { 2 } - 3 t$ have a vertical tangent?

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 5 } { 6 }$ 
E)  $- \frac { 4 } { 3 }$
F) $- \frac { 3 } { 2 }$
G) $- \frac { 2 } { 3 }$
H) $- \frac { 1 } { 3 }$ 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 5 } { 6 }$ 
E)  $- \frac { 4 } { 3 }$
F) $- \frac { 3 } { 2 }$
G) $- \frac { 2 } { 3 }$
H) $- \frac { 1 } { 3 }$

Question 8

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = \left( 9 - \frac { 1 } { x ^ { 2 } } \right) ^ { 2 }$ has zero slope.

Accepted Answer

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

Question 9

If $x ^ { 2 } + y ^ { 2 } = 25$ find the value of $\frac { d y } { d x }$ at the point (3, 4).

Accepted Answer

A)  $\frac { 3 } { 5 }$ 
B)  $- \frac { 3 } { 5 }$ 
C)  $- \frac { 4 } { 5 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 0
F)1
G) $\frac { 4 } { 5 }$ 
H) $- \frac { 3 } { 4 }$ 
A)  $\frac { 3 } { 5 }$ 
B)  $- \frac { 3 } { 5 }$ 
C)  $- \frac { 4 } { 5 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 0
F)1
G) $\frac { 4 } { 5 }$ 
H) $- \frac { 3 } { 4 }$

Question 10

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

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $- \frac { 1 } { 3 }$ 
D) 1 
E)  $\frac { 1 } { 3 }$ 
F) $- \frac { 1 } { 4 }$
G) $\frac { 1 } { 4 }$
H) $- 1$ 
A)  $- \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $- \frac { 1 } { 3 }$ 
D) 1 
E)  $\frac { 1 } { 3 }$ 
F) $- \frac { 1 } { 4 }$
G) $\frac { 1 } { 4 }$
H) $- 1$

Question 11

If $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 1 } , \text { find } \mathrm { f } ^ { \prime \prime } ( 3 )$

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 12 }$ 
C)  $- \frac { 1 } { 12 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 1 } { 3 }$ 
F) $\frac { 4 } { 3 }$
G) $- \frac { 4 } { 3 }$
H)None of these 
A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 12 }$ 
C)  $- \frac { 1 } { 12 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 1 } { 3 }$ 
F) $\frac { 4 } { 3 }$
G) $- \frac { 4 } { 3 }$
H)None of these

Question 12

The population of a bacteria colony after t hours is given by P(t) = $2000 e ^ { 0.087 t }$ . Find the growth rate of the colony when t = 16 hours.

Accepted Answer

A) 0.087 
B) 16 
C) 174 
D) 700 
E) 1600
F)2000
G)20,000
H)32,000 
A) 0.087 
B) 16 
C) 174 
D) 700 
E) 1600
F)2000
G)20,000
H)32,000

Question 13

Find an equation in x and y for the tangent line to the curve $x = e ^ { t } , \mathrm { y } = \mathrm { e } ^ { - t }$ at the point $\left( \frac { 1 } { 3 } , 3 \right)$

Accepted Answer

The answer of Find an equation in x and y...

Question 14

Find an equation of the tangent to the curve $y = \frac { 4 x } { \sqrt { 3 + x ^ { 2 } } }$ at $x = 1$ .

Accepted Answer

The answer of Find an equation of the tangent to...

Question 15

Find the interval on which the graph of $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$ is concave upward.

Accepted Answer

A) (-1, 1) 
B) (-1, 2) 
C) (-2, 1) 
D) (-2, 2) 
E) (-1, 3)
F)(-3, 2)
G)(-3, 3)
H) $( - \infty , \infty )$ 
A) (-1, 1) 
B) (-1, 2) 
C) (-2, 1) 
D) (-2, 2) 
E) (-1, 3)
F)(-3, 2)
G)(-3, 3)
H) $( - \infty , \infty )$

Question 16

Determine the equation of another tangent line to this curve that is parallel to the given tangent line.

Accepted Answer

From the given line, m = 1. No

Question 17

(a) Find the linearization of the function f (x) = sin x when x = 0.(b) Use these results to approximate sin (0.05) and sin (-0.005).

Accepted Answer

(a) l(x) = x
(b) sin

Question 18

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

Accepted Answer

A) 
$\frac { 3 \sqrt { 2 } } { 8 }$ 
B)  $\frac { \sqrt { 2 } } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 8 }$ 
E)  $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { 1 } { 2 }$
G) $\sqrt { 2 }$
H)1 
A) 
$\frac { 3 \sqrt { 2 } } { 8 }$ 
B)  $\frac { \sqrt { 2 } } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 8 }$ 
E)  $\frac { \sqrt { 2 } } { 2 }$
F) $\frac { 1 } { 2 }$
G) $\sqrt { 2 }$
H)1

Question 19

Let $y = x ^ { 4 } + x ^ { 2 } + 1$ , $x = 1$ ,and dx = 1. Find the value of the differential dy.

Accepted Answer

A) 2 
B) 4 
C) 6 
D) 8 
E) 10
F)0.12
G)0
H) $\frac { 1 } { 2 }$ 
A) 2 
B) 4 
C) 6 
D) 8 
E) 10
F)0.12
G)0
H) $\frac { 1 } { 2 }$

Question 20

The relationship between the rate of a certain chemical reaction and temperature under certain circumstances is given by $R ( T ) = 0.1 \left( - 0.05 T ^ { 3 } + 4 T ^ { 2 } + 120 \right)$ grams/sec, where R is the rate of reaction and T is the temperature (in žC).
(a) Find the temperature T at which the reaction rate reaches its maximum.
(b) What is the maximum reaction rate?

Accepted Answer

(a) @#LAT-DLM& So @#LAT-DLM& . Since @#LAT-DLM& an

Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \tan 2 x } { x }$ .

Differentiate the following functions: (a) $f ( x ) = \frac { e ^ { x } } { x }$ (b) $f ( x ) = ( \sqrt { x } + 2 ) ( 2 \sqrt { x } - 4 )$ (c) $f ( x ) = e + 3 x ^ { e }$ (d ) $f ( x ) = \pi ^ { e } \left( x ^ { 2 } - x \right)$

Use implicit differentiation to find $y ^ { \prime \prime } \text { if } 2 x y = y ^ { 2 }$ .

Find the derivative of $f ( t ) = \sqrt { t ^ { 5 } } + \sqrt [ 5 ] { t ^ { 2 } }$ .

Suppose that $h ( x ) = f ( g ( x ) )$ and $g ( 3 ) = 6$ , $g ^ { \prime } ( 3 ) = 4 ,$ $f ^ { \prime } ( 3 ) = 2 , f ^ { \prime } ( 6 ) = 7$ Find the value of $h ^ { \prime } ( 3 ) \text {. }$

Let $f ( x ) = \sin ^ { - 1 } \left( \frac { 4 } { x } \right)$ . Find the value of $f ^ { \prime } ( 8 )$ .

At what value of t does the curve $x = 2 t - 3 t ^ { 2 } , y = t ^ { 2 } - 3 t$ have a vertical tangent?

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = \left( 9 - \frac { 1 } { x ^ { 2 } } \right) ^ { 2 }$ has zero slope.

If $x ^ { 2 } + y ^ { 2 } = 25$ find the value of $\frac { d y } { d x }$ at the point (3, 4).

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

If $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 1 } , \text { find } \mathrm { f } ^ { \prime \prime } ( 3 )$

The population of a bacteria colony after t hours is given by P(t) = $2000 e ^ { 0.087 t }$ . Find the growth rate of the colony when t = 16 hours.

Find an equation in x and y for the tangent line to the curve $x = e ^ { t } , \mathrm { y } = \mathrm { e } ^ { - t }$ at the point $\left( \frac { 1 } { 3 } , 3 \right)$

Find an equation of the tangent to the curve $y = \frac { 4 x } { \sqrt { 3 + x ^ { 2 } } }$ at $x = 1$ .

Find the interval on which the graph of $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$ is concave upward.

Determine the equation of another tangent line to this curve that is parallel to the given tangent line.

(a) Find the linearization of the function f (x) = sin x when x = 0.(b) Use these results to approximate sin (0.05) and sin (-0.005).

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

Let $y = x ^ { 4 } + x ^ { 2 } + 1$ , $x = 1$ ,and dx = 1. Find the value of the differential dy.

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Exam 3: Differentiation Rules

Find the value of the limit lim⁡x→0tan⁡2xx\lim _ { x \rightarrow 0 } \frac { \tan 2 x } { x }limx→0​xtan2x​ .

Use implicit differentiation to find y′′ if 2xy=y2y ^ { \prime \prime } \text { if } 2 x y = y ^ { 2 }y′′ if 2xy=y2 .

Find the derivative of f(t)=t5+t25f ( t ) = \sqrt { t ^ { 5 } } + \sqrt [ 5 ] { t ^ { 2 } }f(t)=t5​+5t2​ .

Suppose that h(x)=f(g(x))h ( x ) = f ( g ( x ) )h(x)=f(g(x)) and g(3)=6g ( 3 ) = 6g(3)=6 , g′(3)=4,g ^ { \prime } ( 3 ) = 4 ,g′(3)=4, f′(3)=2,f′(6)=7f ^ { \prime } ( 3 ) = 2 , f ^ { \prime } ( 6 ) = 7f′(3)=2,f′(6)=7 Find the value of h′(3). h ^ { \prime } ( 3 ) \text {. }h′(3).

Let f(x)=sin⁡−1(4x)f ( x ) = \sin ^ { - 1 } \left( \frac { 4 } { x } \right)f(x)=sin−1(x4​) . Find the value of f′(8)f ^ { \prime } ( 8 )f′(8) .

At what value of t does the curve x=2t−3t2,y=t2−3tx = 2 t - 3 t ^ { 2 } , y = t ^ { 2 } - 3 tx=2t−3t2,y=t2−3t have a vertical tangent?

Find the x-coordinate(s) of the point(s) where the tangent to the curve y=(9−1x2)2y = \left( 9 - \frac { 1 } { x ^ { 2 } } \right) ^ { 2 }y=(9−x21​)2 has zero slope.

If x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25 find the value of dydx\frac { d y } { d x }dxdy​ at the point (3, 4).

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

If f(x)=x2−13, find f′′(3)f ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 1 } , \text { find } \mathrm { f } ^ { \prime \prime } ( 3 )f(x)=3x2−1​, find f′′(3)

The population of a bacteria colony after t hours is given by P(t) = 2000e0.087t2000 e ^ { 0.087 t }2000e0.087t . Find the growth rate of the colony when t = 16 hours.

Find an equation in x and y for the tangent line to the curve x=et,y=e−tx = e ^ { t } , \mathrm { y } = \mathrm { e } ^ { - t }x=et,y=e−t at the point (13,3)\left( \frac { 1 } { 3 } , 3 \right)(31​,3)

Find an equation of the tangent to the curve y=4x3+x2y = \frac { 4 x } { \sqrt { 3 + x ^ { 2 } } }y=3+x2​4x​ at x=1x = 1x=1 .

Find the interval on which the graph of f(x)=ln⁡(x2+1)f ( x ) = \ln \left( x ^ { 2 } + 1 \right)f(x)=ln(x2+1) is concave upward.

Determine the equation of another tangent line to this curve that is parallel to the given tangent line.

(a) Find the linearization of the function f (x) = sin x when x = 0.(b) Use these results to approximate sin (0.05) and sin (-0.005).

If f(x)=x+x, find f′(1)f ( x ) = \sqrt { x + \sqrt { x } } , \text { find } f ^ { \prime } ( 1 )f(x)=x+x​​, find f′(1)

Let y=x4+x2+1y = x ^ { 4 } + x ^ { 2 } + 1y=x4+x2+1 , x=1x = 1x=1 ,and dx = 1. Find the value of the differential dy.

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Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \tan 2 x } { x }$ .

Use implicit differentiation to find $y ^ { \prime \prime } \text { if } 2 x y = y ^ { 2 }$ .

Find the derivative of $f ( t ) = \sqrt { t ^ { 5 } } + \sqrt [ 5 ] { t ^ { 2 } }$ .

Suppose that $h ( x ) = f ( g ( x ) )$ and $g ( 3 ) = 6$ , $g ^ { \prime } ( 3 ) = 4 ,$ $f ^ { \prime } ( 3 ) = 2 , f ^ { \prime } ( 6 ) = 7$ Find the value of $h ^ { \prime } ( 3 ) \text {. }$

Let $f ( x ) = \sin ^ { - 1 } \left( \frac { 4 } { x } \right)$ . Find the value of $f ^ { \prime } ( 8 )$ .

At what value of t does the curve $x = 2 t - 3 t ^ { 2 } , y = t ^ { 2 } - 3 t$ have a vertical tangent?

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = \left( 9 - \frac { 1 } { x ^ { 2 } } \right) ^ { 2 }$ has zero slope.

If $x ^ { 2 } + y ^ { 2 } = 25$ find the value of $\frac { d y } { d x }$ at the point (3, 4).

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

If $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 1 } , \text { find } \mathrm { f } ^ { \prime \prime } ( 3 )$

The population of a bacteria colony after t hours is given by P(t) = $2000 e ^ { 0.087 t }$ . Find the growth rate of the colony when t = 16 hours.

Find an equation in x and y for the tangent line to the curve $x = e ^ { t } , \mathrm { y } = \mathrm { e } ^ { - t }$ at the point $\left( \frac { 1 } { 3 } , 3 \right)$

Find an equation of the tangent to the curve $y = \frac { 4 x } { \sqrt { 3 + x ^ { 2 } } }$ at $x = 1$ .

Find the interval on which the graph of $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$ is concave upward.

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

Let $y = x ^ { 4 } + x ^ { 2 } + 1$ , $x = 1$ ,and dx = 1. Find the value of the differential dy.