Question 1

If $f ( x ) = - 2 ^ { - b x } , \text { find } f ^ { ( 10 ) } ( x )$ where k is a constant.

Accepted Answer

A)  $2 ^ { - k x } ( - k \ln 2 )$ 
B)  $2 ^ { - k x } ( k \ln 2 )$ 
C)  $2 ^ { - k x } \left( - k ^ { 10 } \ln 2 \right)$ 
D)  $2 ^ { - k x } \left( k ^ { 10 } \ln 2 \right)$ 
E)  $2 ^ { - 6 x } \left( k ^ { 10 } ( \ln 2 ) ^ { 10 } \right)$ 
F) $2 ^ { - 5 x } k ^ { 10 } 2 ^ { 10 }$ 
G) $2 ^ { - k x } \left( - k ^ { 10 } \right) 2 ^ { 10 }$ 
H) $2 ^ { - k x } ( - 2 k )$ 
A)  $2 ^ { - k x } ( - k \ln 2 )$ 
B)  $2 ^ { - k x } ( k \ln 2 )$ 
C)  $2 ^ { - k x } \left( - k ^ { 10 } \ln 2 \right)$ 
D)  $2 ^ { - k x } \left( k ^ { 10 } \ln 2 \right)$ 
E)  $2 ^ { - 6 x } \left( k ^ { 10 } ( \ln 2 ) ^ { 10 } \right)$ 
F) $2 ^ { - 5 x } k ^ { 10 } 2 ^ { 10 }$ 
G) $2 ^ { - k x } \left( - k ^ { 10 } \right) 2 ^ { 10 }$ 
H) $2 ^ { - k x } ( - 2 k )$

Question 2

(a) Differentiate $e ^ { 2 x }$ by differentiating $e ^ { x } \cdot e ^ { x }$ .(b) Differentiate $e ^ { 3 x }$ by differentiating $e ^ { x } e ^ { 2 x }$ and using the result of part (a).(c) Continue as above to find $\frac { d } { d x } e ^ { 4 x } \text { and } \frac { d } { d x } e ^ { 5 x }$ using the results from above.(d) Based upon your answers to parts (a)-(c), make a conjecture about $\frac { d } { d x } e ^ { n x }$ .

Accepted Answer

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

Question 3

Below is a table of the vapor pressure (in kilopascals) of water for various temperatures (in degrees Kelvin): $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Pressure } ( \mathrm { kPa } ) & 4.6 & 9.2 & 17.5 & 31.8 & 55.3 & 92.5 & 149.4 & 233.7 & 355.1 & 525.8 & 760 \
\hline \text { Temperature } \left( { } ^ { \circ } \mathrm { K } \right) & 273 & 283 & 293 & 303 & 313 & 323 & 333 & 343 & 353 & 363 & 373 \
\hline
\end{array}$$ (a) Estimate the rate of change of pressure with respect to temperature on the following intervals:
(i) [363, 373]
(ii) [333, 343]
(iii) [273, 283]
(b) Plot the points from the table and fit an appropriate exponential model to these data.(c) From the model in part (b), determine the instantaneous rate of change of pressure with respect to temperature.(d) Is the rate of change of pressure increasing or decreasing with respect to temperature? Justify your answer.

Accepted Answer

(a) (i) 23.42
(ii) 8.43
(iii)

Question 4

Show that the curve $y = x ^ { 3 } + 2 x + \cos x$ has no tangent line with slope 0.

Accepted Answer

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

Question 5

Use differentials to approximate the change in the function $f ( x ) = x ^ { 2 } + x - 2$ when x varies from 1 to 1.01.

Accepted Answer

@#LAT-DLM&y @#LAT-DLM& dy = f'(1)@#LAT-DLM&x = (2x

Question 6

The position of a particle is given by the function s(t) = $2 t ^ { 3 } - 9 t ^ { 2 } + 12 t$ , where t is measured in seconds and s in meters.(a) Find the velocity at time t.(b) When is the particle at rest?
(c) When is the particle moving in the positive direction?
(d) Draw a diagram to represent the motion of the particle.(e) Find the total distance traveled by the particle during the time interval [1,3].

Accepted Answer

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

Question 7

Find the derivative $\frac { d p } { d t }$ if $p = \frac { m v } { \sqrt { 1 - \left( v ^ { 2 } / c ^ { 2 } \right) } }$ where m and c are constants, v is velocity function.

Accepted Answer

The answer of Find the derivative \(\frac { d p...

Question 8

If $y = x ^ { 2 } e ^ { x } , \text { find } f ^ { \prime \prime } ( 0 )$

Accepted Answer

A)  $- 2$ 
B) 0 
C) 2 
D) 4 
E) 2e
F)4e
G)6e
H)7e 
A)  $- 2$ 
B) 0 
C) 2 
D) 4 
E) 2e
F)4e
G)6e
H)7e

Question 9

Find an equation of the tangent line to the curve $y = e ^ { x } ( x + 1 )$ at $x = 0$ .

Accepted Answer

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

Question 10

Lake bottoms are frequently mapped using contour lines, which are curves joining points of
the same depth. The path of steepest descent is orthogonal to the contour lines. Given the
contour map below, sketch the path of steepest descent from starting positions A and B to
the deepest point C.

Accepted Answer

The answer of Lake bottoms are frequently mapped using contour...

Question 11

Each of the following is a derivative of a function obtained by using the Quotient Rule. Determine the original function:
(a) $f ^ { \prime } ( x ) = \frac { ( x + 3 ) - ( x + 1 ) } { ( x + 3 ) ^ { 2 } }$ (b) $g ^ { \prime } ( x ) = \frac { 3 x ^ { 2 } e ^ { x } - x ^ { 3 } e ^ { x } } { e ^ { 2 x } }$ (c) $h ^ { \prime } ( x ) = \frac { x ^ { 3 } e ^ { x } - 3 x ^ { 2 } e ^ { x } } { x ^ { 6 } }$

Accepted Answer

The answer of Each of the following is a derivative...

Question 12

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

Accepted Answer

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

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

F)4
G) $\frac { 1 } { 4 }$ 
H)3

Question 13

If $f ( x ) = e ^ { x } + x ^ { e } + e ^ { x }$ , what are $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ ?

Accepted Answer

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

Question 14

Find an equation of the tangent line to the given curve at the given point.(a) $y = \frac { \sin x } { x + 1 } \text { at } ( 0,0 )$ (b) $y = \frac { \cos x } { x + 1 } \text { at } \left( 0 , \frac { 1 } { 2 } \right)$

Accepted Answer

(a) @#IMG-DLM& so the slope of the tange

Question 15

Let $p ( t ) = \frac { 1,000 \left( e ^ { 0.1 t } + 3 \right) } { e ^ { 0.1 t } + 9 }$ 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 16

Given $f ( x ) = \sqrt { x }$ (a) Find an equation of the tangent line to the graph of $y = f ( x )$ at
(i) x = 0.(ii) x = 4.(iii) x = 9.(b) Sketch a graph of $y = f ( x )$ and the tangent lines you found in parts (i), (ii) and (iii) on one set of coordinate axes.

Accepted Answer

(a) @#IMG-DLM& (i) The slope of the tangent is und

Question 17

If $f ( x ) = \frac { x } { \tan x } , \text { find } f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

Accepted Answer

A)  $\frac { 2 - \pi } { 2 }$ 
B)  $\frac { 1 - \pi } { 2 }$ 
C)  $1 - \pi$ 
D)  $\frac { \pi } { 2 }$ 
E)  $1 - 2 \pi$
F) $2 - \pi$

G) $2 - 2 \pi$

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

G) $2 - 2 \pi$

H) $- \pi$

Question 18

Find $\frac { d y } { d x }$ if $x y + e ^ { 3 x } = \sin ( x + y )$ .

Accepted Answer

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

Question 19

Find the slope of the tangent to the curve with parametric equations $x = t + t ^ { 2 } , y = t + e ^ { t }$ at the point (0, 1).

Accepted Answer

A)  $- 3$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)3
H)4 
A)  $- 3$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)3
H)4

Question 20

Find the derivative of $f ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 4 } , \text { find } \mathrm { f } ^ { \prime } ( 0 )$

Accepted Answer

A) 0 
B) 28 
C) 4 
D) 32 
E) 16
F)24
G)12
H)8 
A) 0 
B) 28 
C) 4 
D) 32 
E) 16
F)24
G)12
H)8

If $f ( x ) = - 2 ^ { - b x } , \text { find } f ^ { ( 10 ) } ( x )$ where k is a constant.

Show that the curve $y = x ^ { 3 } + 2 x + \cos x$ has no tangent line with slope 0.

Use differentials to approximate the change in the function $f ( x ) = x ^ { 2 } + x - 2$ when x varies from 1 to 1.01.

Find the derivative $\frac { d p } { d t }$ if $p = \frac { m v } { \sqrt { 1 - \left( v ^ { 2 } / c ^ { 2 } \right) } }$ where m and c are constants, v is velocity function.

If $y = x ^ { 2 } e ^ { x } , \text { find } f ^ { \prime \prime } ( 0 )$

Find an equation of the tangent line to the curve $y = e ^ { x } ( x + 1 )$ at $x = 0$ .

Lake bottoms are frequently mapped using contour lines, which are curves joining points of the same depth. The path of steepest descent is orthogonal to the contour lines. Given the contour map below, sketch the path of steepest descent from starting positions A and B to the deepest point C.

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

If $f ( x ) = e ^ { x } + x ^ { e } + e ^ { x }$ , what are $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ ?

Find an equation of the tangent line to the given curve at the given point.(a) $y = \frac { \sin x } { x + 1 } \text { at } ( 0,0 )$ (b) $y = \frac { \cos x } { x + 1 } \text { at } \left( 0 , \frac { 1 } { 2 } \right)$

Let $p ( t ) = \frac { 1,000 \left( e ^ { 0.1 t } + 3 \right) } { e ^ { 0.1 t } + 9 }$ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

Given $f ( x ) = \sqrt { x }$ (a) Find an equation of the tangent line to the graph of $y = f ( x )$ at (i) x = 0.(ii) x = 4.(iii) x = 9.(b) Sketch a graph of $y = f ( x )$ and the tangent lines you found in parts (i), (ii) and (iii) on one set of coordinate axes.

If $f ( x ) = \frac { x } { \tan x } , \text { find } f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

Find $\frac { d y } { d x }$ if $x y + e ^ { 3 x } = \sin ( x + y )$ .

Find the slope of the tangent to the curve with parametric equations $x = t + t ^ { 2 } , y = t + e ^ { t }$ at the point (0, 1).

Find the derivative of $f ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 4 } , \text { find } \mathrm { f } ^ { \prime } ( 0 )$

Functions and Models

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Exam 3: Differentiation Rules

If f(x)=−2−bx, find f(10)(x)f ( x ) = - 2 ^ { - b x } , \text { find } f ^ { ( 10 ) } ( x )f(x)=−2−bx, find f(10)(x) where k is a constant.

Show that the curve y=x3+2x+cos⁡xy = x ^ { 3 } + 2 x + \cos xy=x3+2x+cosx has no tangent line with slope 0.

Use differentials to approximate the change in the function f(x)=x2+x−2f ( x ) = x ^ { 2 } + x - 2f(x)=x2+x−2 when x varies from 1 to 1.01.

Find the derivative dpdt\frac { d p } { d t }dtdp​ if p=mv1−(v2/c2)p = \frac { m v } { \sqrt { 1 - \left( v ^ { 2 } / c ^ { 2 } \right) } }p=1−(v2/c2)​mv​ where m and c are constants, v is velocity function.

If y=x2ex, find f′′(0)y = x ^ { 2 } e ^ { x } , \text { find } f ^ { \prime \prime } ( 0 )y=x2ex, find f′′(0)

Find an equation of the tangent line to the curve y=ex(x+1)y = e ^ { x } ( x + 1 )y=ex(x+1) at x=0x = 0x=0 .

Lake bottoms are frequently mapped using contour lines, which are curves joining points of the same depth. The path of steepest descent is orthogonal to the contour lines. Given the contour map below, sketch the path of steepest descent from starting positions A and B to the deepest point C.

If f(x)=x2cos⁡x, find f′(0)f ( x ) = x ^ { 2 } \cos x , \text { find } f ^ { \prime } ( 0 )f(x)=x2cosx, find f′(0)

If f(x)=ex+xe+exf ( x ) = e ^ { x } + x ^ { e } + e ^ { x }f(x)=ex+xe+ex , what are f′(x)f ^ { \prime } ( x )f′(x) and f′′(x)f ^ { \prime \prime } ( x )f′′(x) ?

Let p(t)=1,000(e0.1t+3)e0.1t+9p ( t ) = \frac { 1,000 \left( e ^ { 0.1 t } + 3 \right) } { e ^ { 0.1 t } + 9 }p(t)=e0.1t+91,000(e0.1t+3)​ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

Given f(x)=xf ( x ) = \sqrt { x }f(x)=x​ (a) Find an equation of the tangent line to the graph of y=f(x)y = f ( x )y=f(x) at (i) x = 0.(ii) x = 4.(iii) x = 9.(b) Sketch a graph of y=f(x)y = f ( x )y=f(x) and the tangent lines you found in parts (i), (ii) and (iii) on one set of coordinate axes.

If f(x)=xtan⁡x, find f′(π4)f ( x ) = \frac { x } { \tan x } , \text { find } f ^ { \prime } \left( \frac { \pi } { 4 } \right)f(x)=tanxx​, find f′(4π​) .

Find dydx\frac { d y } { d x }dxdy​ if xy+e3x=sin⁡(x+y)x y + e ^ { 3 x } = \sin ( x + y )xy+e3x=sin(x+y) .

Find the slope of the tangent to the curve with parametric equations x=t+t2,y=t+etx = t + t ^ { 2 } , y = t + e ^ { t }x=t+t2,y=t+et at the point (0, 1).

Find the derivative of f(x)=(x2+1)4, find f′(0)f ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 4 } , \text { find } \mathrm { f } ^ { \prime } ( 0 )f(x)=(x2+1)4, find f′(0)

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 $f ( x ) = - 2 ^ { - b x } , \text { find } f ^ { ( 10 ) } ( x )$ where k is a constant.

Show that the curve $y = x ^ { 3 } + 2 x + \cos x$ has no tangent line with slope 0.

Use differentials to approximate the change in the function $f ( x ) = x ^ { 2 } + x - 2$ when x varies from 1 to 1.01.

Find the derivative $\frac { d p } { d t }$ if $p = \frac { m v } { \sqrt { 1 - \left( v ^ { 2 } / c ^ { 2 } \right) } }$ where m and c are constants, v is velocity function.

If $y = x ^ { 2 } e ^ { x } , \text { find } f ^ { \prime \prime } ( 0 )$

Find an equation of the tangent line to the curve $y = e ^ { x } ( x + 1 )$ at $x = 0$ .

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

If $f ( x ) = e ^ { x } + x ^ { e } + e ^ { x }$ , what are $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ ?

Let $p ( t ) = \frac { 1,000 \left( e ^ { 0.1 t } + 3 \right) } { e ^ { 0.1 t } + 9 }$ be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria after 10 hours.

Given $f ( x ) = \sqrt { x }$ (a) Find an equation of the tangent line to the graph of $y = f ( x )$ at (i) x = 0.(ii) x = 4.(iii) x = 9.(b) Sketch a graph of $y = f ( x )$ and the tangent lines you found in parts (i), (ii) and (iii) on one set of coordinate axes.

If $f ( x ) = \frac { x } { \tan x } , \text { find } f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

Find $\frac { d y } { d x }$ if $x y + e ^ { 3 x } = \sin ( x + y )$ .

Find the slope of the tangent to the curve with parametric equations $x = t + t ^ { 2 } , y = t + e ^ { t }$ at the point (0, 1).

Find the derivative of $f ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 4 } , \text { find } \mathrm { f } ^ { \prime } ( 0 )$