Question 1

If $f ( x ) = 2 e ^ { x } - 3 x ^ { e } + \sqrt { e }$ , find $f ^ { \prime } ( x )$ .

Accepted Answer

A)  $2 x e ^ { x - 1 } - 3 x e ^ { e - 1 }$ 
B)  $2 e ^ { x } - 3 x ^ { e } + \sqrt { e }$ 
C)  $2 x e ^ { x - 1 } - 3 x ^ { e - 1 } + \frac { 1 } { 2 } e ^ { - 1 / 2 }$ 
D)  $2 e ^ { x } - 3 e x ^ { e - 1 }$ 
E)  $e ^ { x } - 3 e x ^ { e - 1 }$ 
F) $2 e ^ { x } - 3 x ^ { e - 1 }$

G) $2 e ^ { x } - 3 x ^ { e }$

H) $2 e ^ { x } - e x ^ { e - 1 }$ 
A)  $2 x e ^ { x - 1 } - 3 x e ^ { e - 1 }$ 
B)  $2 e ^ { x } - 3 x ^ { e } + \sqrt { e }$ 
C)  $2 x e ^ { x - 1 } - 3 x ^ { e - 1 } + \frac { 1 } { 2 } e ^ { - 1 / 2 }$ 
D)  $2 e ^ { x } - 3 e x ^ { e - 1 }$ 
E)  $e ^ { x } - 3 e x ^ { e - 1 }$ 
F) $2 e ^ { x } - 3 x ^ { e - 1 }$

G) $2 e ^ { x } - 3 x ^ { e }$

H) $2 e ^ { x } - e x ^ { e - 1 }$

Question 2

The quantity q of Zeng Athletic Shoes which are sold depends on the selling price p. [That is, q = f(p).]
(a) If you know that f (150) = 14,000, what can you say about the sale of these shoes?
(b) If you know that $f ^ { \prime } ( 150 ) = - 100$ , what does that tell you about the sale of these shoes?
(c) The total revenue, R, earned through the sale of Zeng shoes is given by $R = p \cdot q$ . Find $\frac { d R } { d p } \text { when } p = 150$ (d) Suppose that the shoes are currently priced at $150. What effect will lowering the price likely have on the total revenue? Justify your answer.

Accepted Answer

(a) 14,000 pairs of shoes would be sold

Question 3

Let $F ( x ) = \sin ( g ( x ) )$ where g is differentiable. Find $F ^ { \prime } ( x )$

Accepted Answer

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

Question 4

Find the point where the tangent to the curve $y = x \sqrt { 3 - x }$ has zero slope.

Accepted Answer

The answer of Find the point where the tangent to...

Question 5

If $f ( x ) = 1 / x ^ { 2 }$ , find $f ^ { \prime \prime } ( 1 )$ .

Accepted Answer

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

Question 6

Consider the curve given by $x = t ^ { 2 } + 3 , y = 2 t ^ { 3 } - t$ Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point corresponding to $t = 2$

Accepted Answer

The answer of Consider the curve given by \(x =...

Question 7

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

Accepted Answer

A) 1.8 
B) 0.4 
C)  $- 0.9$ 
D)  $- 1.8$ 
E) 0.9
F) $- 0.4$
G)0.77
H) $- 0.77$ 
A) 1.8 
B) 0.4 
C)  $- 0.9$ 
D)  $- 1.8$ 
E) 0.9
F) $- 0.4$
G)0.77
H) $- 0.77$

Question 8

Show that the curves $3 x ^ { 2 } + 2 x - 3 y ^ { 2 } = 1$ and $6 x y + 2 y = 0$ are orthogonal.

Accepted Answer

@#IMG-DLM& @#IMG-DLM& , 6xy +

Question 9

There are two lines passing through the point (1, 0) tangent to the parabola $y ^ { 2 } + 4 x = 0$ . Find their equations.

Accepted Answer

y = -x + 1

Question 10

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

Accepted Answer

A)  $\frac { 2 t ^ { 2 } + 3 } { t ^ { 2 } }$ 
B)  $\frac { 2 t ^ { 2 } - 3 } { t ^ { 2 } }$ 
C)  $3 t ^ { 2 }$ 
D)  $\frac { 2 t ^ { 3 } - 3 } { t ^ { 2 } }$ 
E)  $3 t ^ { 2 } + 3$
F) $\frac { 2 t ^ { 3 } + 3 } { t ^ { 2 } }$

G) $\frac { 2 t ^ { 3 } + 3 } { t }$

H)None of the above 
A)  $\frac { 2 t ^ { 2 } + 3 } { t ^ { 2 } }$ 
B)  $\frac { 2 t ^ { 2 } - 3 } { t ^ { 2 } }$ 
C)  $3 t ^ { 2 }$ 
D)  $\frac { 2 t ^ { 3 } - 3 } { t ^ { 2 } }$ 
E)  $3 t ^ { 2 } + 3$
F) $\frac { 2 t ^ { 3 } + 3 } { t ^ { 2 } }$

G) $\frac { 2 t ^ { 3 } + 3 } { t }$

H)None of the above

Question 11

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

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B) 3 
C) 0 
D) 1 
E)  $\frac { 1 } { 2 }$

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

F) $\frac { 1 } { 4 }$ 
G)4
H)2

Question 12

Suppose that u and v are differentiable functions and that $w = u \circ v$ and $u ( 0 ) = 1 ,$ $v ( 0 ) = 2$ $u ^ { \prime } ( 0 ) = 3 , u ^ { \prime } ( 2 ) = 4 , v ^ { \prime } ( 0 ) = 5 , v ^ { \prime } ( 2 ) = 6$ Find $w ^ { \prime } ( 0 )$

Accepted Answer

The answer of Suppose that u and v are differentiable...

Question 13

If $f ( x ) = \sqrt { 3 x }$ , find $f ^ { \prime } ( 3 )$

Accepted Answer

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

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

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

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

H) $\frac { 1 } { 6 }$

Question 14

Find the derivative of $f ( x ) = \sqrt { 9 x }$ .

Accepted Answer

A)  $\frac { 3 } { 2 \sqrt { x } }$ 
B)  $- \frac { 3 } { 2 \sqrt { x } }$ 
C)  $\frac { 9 } { 2 \sqrt { x } }$ 
D)  $- \frac { 9 } { 2 \sqrt { x } }$ 
E)  $\frac { 3 } { \sqrt { x } }$
F) $- 3 \sqrt { x }$

G) $3 \sqrt { x }$

H)None of the above 
A)  $\frac { 3 } { 2 \sqrt { x } }$ 
B)  $- \frac { 3 } { 2 \sqrt { x } }$ 
C)  $\frac { 9 } { 2 \sqrt { x } }$ 
D)  $- \frac { 9 } { 2 \sqrt { x } }$ 
E)  $\frac { 3 } { \sqrt { x } }$
F) $- 3 \sqrt { x }$

G) $3 \sqrt { x }$

H)None of the above

Question 15

Profit (in dollars) for a company when x units of a certain product are produced is given by $P ( x ) = ( 100 - x ) \ln x$ when x > 1.(a) What is the marginal profit (the derivative of the profit function)?
(b) If the current production level is x = 15, is the profit increasing or decreasing?
(c) If the current production level is x = 40, is the profit increasing or decreasing?
(d) At approximately what production level does the profit function reach its maximum value? What is the maximum profit?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& so the profit is inc

Question 16

Suppose you are given a function f where $f ( 4 ) = 5 , f ^ { \prime } ( 4 ) = - 2 \text { and } f ^ { \prime \prime } ( 4 ) = 3 \text {. If } F ( x ) = \sqrt { x } f ( x )$ determine:
(a) $F ^ { \prime } ( 4 )$ (b) $F ^ { \prime \prime } ( 4 )$

Accepted Answer

The answer of Suppose you are given a function f...

Question 17

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

Accepted Answer

A) 2 
B)  $e ^ { 2 }$ 
C) ln $\left( \frac { 1 } { 2 } \right)$ 
D)  $e ^ { - \frac { 1 } { 2 } }$ 
E)  $2 ^ { e }$
F) $\frac { 1 } { 2 }$
G) $\frac { 1 } { \ln 2 }$
H) $e ^ { \frac { 1 } { 2 } }$ 
A) 2 
B)  $e ^ { 2 }$ 
C) ln $\left( \frac { 1 } { 2 } \right)$ 
D)  $e ^ { - \frac { 1 } { 2 } }$ 
E)  $2 ^ { e }$
F) $\frac { 1 } { 2 }$
G) $\frac { 1 } { \ln 2 }$
H) $e ^ { \frac { 1 } { 2 } }$

Question 18

Given $f ( x ) = \frac { x - 4 } { x - 1 }$ , find an equation of the line(s) tangent to the graph of $y = f ( x )$ and parallel to $y - 3 x + 7 = 0$ .

Accepted Answer

@#IMG-DLM& , so @#IMG-DLM& . The tangent lines mus

Question 19

The position of a particle moving along the x-axis is given by $x = 0.08 \sin ( 12 t + 0.3 )$ meters, where t is measured in seconds.(a) Determine the position, velocity, and acceleration of the particle when t = 0.65.(b) Show that the acceleration of the particle is proportional to its position, but in the opposite direction.

Accepted Answer

The answer of The position of a particle moving along...

Question 20

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

Accepted Answer

A) 2 
B) 4 
C) 6 
D) 8 
E)  $e ^ { - 1 }$ 
F) $2 e ^ { - 1 }$ 
G) $4 e ^ { - 1 }$ 
H) $6 e ^ { - 1 }$ 
A) 2 
B) 4 
C) 6 
D) 8 
E)  $e ^ { - 1 }$ 
F) $2 e ^ { - 1 }$ 
G) $4 e ^ { - 1 }$ 
H) $6 e ^ { - 1 }$

If $f ( x ) = 2 e ^ { x } - 3 x ^ { e } + \sqrt { e }$ , find $f ^ { \prime } ( x )$ .

Let $F ( x ) = \sin ( g ( x ) )$ where g is differentiable. Find $F ^ { \prime } ( x )$

Find the point where the tangent to the curve $y = x \sqrt { 3 - x }$ has zero slope.

If $f ( x ) = 1 / x ^ { 2 }$ , find $f ^ { \prime \prime } ( 1 )$ .

Consider the curve given by $x = t ^ { 2 } + 3 , y = 2 t ^ { 3 } - t$ Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point corresponding to $t = 2$

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

Show that the curves $3 x ^ { 2 } + 2 x - 3 y ^ { 2 } = 1$ and $6 x y + 2 y = 0$ are orthogonal.

There are two lines passing through the point (1, 0) tangent to the parabola $y ^ { 2 } + 4 x = 0$ . Find their equations.

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

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

Suppose that u and v are differentiable functions and that $w = u \circ v$ and $u ( 0 ) = 1 ,$ $v ( 0 ) = 2$ $u ^ { \prime } ( 0 ) = 3 , u ^ { \prime } ( 2 ) = 4 , v ^ { \prime } ( 0 ) = 5 , v ^ { \prime } ( 2 ) = 6$ Find $w ^ { \prime } ( 0 )$

If $f ( x ) = \sqrt { 3 x }$ , find $f ^ { \prime } ( 3 )$

Find the derivative of $f ( x ) = \sqrt { 9 x }$ .

Suppose you are given a function f where $f ( 4 ) = 5 , f ^ { \prime } ( 4 ) = - 2 \text { and } f ^ { \prime \prime } ( 4 ) = 3 \text {. If } F ( x ) = \sqrt { x } f ( x )$ determine: (a) $F ^ { \prime } ( 4 )$ (b) $F ^ { \prime \prime } ( 4 )$

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

Given $f ( x ) = \frac { x - 4 } { x - 1 }$ , find an equation of the line(s) tangent to the graph of $y = f ( x )$ and parallel to $y - 3 x + 7 = 0$ .

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

Functions and Models

Limits and Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Exam 3: Differentiation Rules

If f(x)=2ex−3xe+ef ( x ) = 2 e ^ { x } - 3 x ^ { e } + \sqrt { e }f(x)=2ex−3xe+e​ , find f′(x)f ^ { \prime } ( x )f′(x) .

Let F(x)=sin⁡(g(x))F ( x ) = \sin ( g ( x ) )F(x)=sin(g(x)) where g is differentiable. Find F′(x)F ^ { \prime } ( x )F′(x)

Find the point where the tangent to the curve y=x3−xy = x \sqrt { 3 - x }y=x3−x​ has zero slope.

If f(x)=1/x2f ( x ) = 1 / x ^ { 2 }f(x)=1/x2 , find f′′(1)f ^ { \prime \prime } ( 1 )f′′(1) .

Consider the curve given by x=t2+3,y=2t3−tx = t ^ { 2 } + 3 , y = 2 t ^ { 3 } - tx=t2+3,y=2t3−t Find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ at the point corresponding to t=2t = 2t=2

Show that the curves 3x2+2x−3y2=13 x ^ { 2 } + 2 x - 3 y ^ { 2 } = 13x2+2x−3y2=1 and 6xy+2y=06 x y + 2 y = 06xy+2y=0 are orthogonal.

There are two lines passing through the point (1, 0) tangent to the parabola y2+4x=0y ^ { 2 } + 4 x = 0y2+4x=0 . Find their equations.

Find the derivative of f(t)=t3+3tf ( t ) = \frac { t ^ { 3 } + 3 } { t }f(t)=tt3+3​ .

Find the value of the limit lim⁡x→0sin⁡x2x\lim _ { x \rightarrow 0 } \frac { \sin x } { 2 x }limx→0​2xsinx​ .

If f(x)=3xf ( x ) = \sqrt { 3 x }f(x)=3x​ , find f′(3)f ^ { \prime } ( 3 )f′(3)

Find the derivative of f(x)=9xf ( x ) = \sqrt { 9 x }f(x)=9x​ .

Let f(x)=log⁡2xf ( x ) = \log _ { 2 } xf(x)=log2​x . Find the value of f′(1)f ^ { \prime } ( 1 )f′(1) .

Given f(x)=x−4x−1f ( x ) = \frac { x - 4 } { x - 1 }f(x)=x−1x−4​ , find an equation of the line(s) tangent to the graph of y=f(x)y = f ( x )y=f(x) and parallel to y−3x+7=0y - 3 x + 7 = 0y−3x+7=0 .

Let f(x)=e−x2f ( x ) = e ^ { - x ^ { 2 } }f(x)=e−x2 Find the value of f′′(1)f ^ { \prime \prime } ( 1 )f′′(1)

Functions and Models

Limits and Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

If $f ( x ) = 2 e ^ { x } - 3 x ^ { e } + \sqrt { e }$ , find $f ^ { \prime } ( x )$ .

Let $F ( x ) = \sin ( g ( x ) )$ where g is differentiable. Find $F ^ { \prime } ( x )$

Find the point where the tangent to the curve $y = x \sqrt { 3 - x }$ has zero slope.

If $f ( x ) = 1 / x ^ { 2 }$ , find $f ^ { \prime \prime } ( 1 )$ .

Consider the curve given by $x = t ^ { 2 } + 3 , y = 2 t ^ { 3 } - t$ Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point corresponding to $t = 2$

Show that the curves $3 x ^ { 2 } + 2 x - 3 y ^ { 2 } = 1$ and $6 x y + 2 y = 0$ are orthogonal.

There are two lines passing through the point (1, 0) tangent to the parabola $y ^ { 2 } + 4 x = 0$ . Find their equations.

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

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

If $f ( x ) = \sqrt { 3 x }$ , find $f ^ { \prime } ( 3 )$

Find the derivative of $f ( x ) = \sqrt { 9 x }$ .

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

Given $f ( x ) = \frac { x - 4 } { x - 1 }$ , find an equation of the line(s) tangent to the graph of $y = f ( x )$ and parallel to $y - 3 x + 7 = 0$ .

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