Question 1

Determine the number of solutions of the equation 5x $4$ - 3x $2$ = 10x $5$ - 20x $6$ $+$ 63x $+$ 5.

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7

Question 2

The following time-of-day and temperature (FS1U1P1$\circ$S1S1P0) were gathered during a gorgeous midsummer day in Fargo, North Dakota:
$$\begin{array} { | c | c | } 
\hline \text { Time of Day } & \text { Temperature } \
\hline 18 & 74 \
\hline 17 & 73 \
\hline 16 & 73 \
\hline 15 & 72 \
\hline 14 & 70 \
\hline 13 & 70 \
\hline 12 & 68 \
\hline 11 & 66 \
\hline 10 & 63 \
\hline 9 & 62 \
\hline 8 & 59 \
\hline 7 & 58 \
\hline
\end{array}$$
Source: National Weather Service; www.weather.gov
(a) Make a scatter plot of these data.
(b) Fit a linear model to the data.
(c) Fit an exponential model to the data.
(d) Fit a quadratic model to the data.(e) Use your equations to make a table showing the predicted temperature for each model, rounded to the nearest degree.
(f) The actual temperature at 8:00 p.m. (20 hours) was 70° F. Which model was closest? Which model was second-closest?
(g) All of the models give values that are too high for each of the times after 6:00 PM. What is one possible explanation for this?

Accepted Answer

@#IMG-DLM& 
(b) y = 1.561x @#LAT-DLM& 47.68
(c) y = 49.89802

Question 3

Find the domain of the inverse for f (x) = $\sqrt { 2 x - 5 }$ .

Accepted Answer

A) ( $- \infty , - \frac { 5 } { 2 }$ ] 
B) ( $- \infty$ , 0] 
C)  $\left[ - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right]$ 
D)  $\left( - \infty , \frac { 5 } { 2 } \right]$ 
E)  $\left[ - \frac { 5 } { 2 } , \infty \right)$ 
F)[0, $\infty$ )

G) $\left[ \frac { 2 } { 5 } , \infty \right)$ 
H) $\left[ \frac { 5 } { 2 } , \infty \right)$ 
A) ( $- \infty , - \frac { 5 } { 2 }$ ] 
B) ( $- \infty$ , 0] 
C)  $\left[ - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right]$ 
D)  $\left( - \infty , \frac { 5 } { 2 } \right]$ 
E)  $\left[ - \frac { 5 } { 2 } , \infty \right)$ 
F)[0, $\infty$ )

G) $\left[ \frac { 2 } { 5 } , \infty \right)$ 
H) $\left[ \frac { 5 } { 2 } , \infty \right)$

Question 4

Find functions f and g such that F(x) = 1 $\sqrt { 1 - \cos ^ { 2 } x }$ = $( f \circ g ) ( x )$

Accepted Answer

f (x) = @#IMG-DLM& , g (x) = 1

Question 5

Which of the following are graphs of functions?

Accepted Answer

A) I only 
B) II only 
C) III only 
D) I and II only 
E) I and III only
F)I, II, and IV only
G)II and V only
H)I,II, III only 
A) I only 
B) II only 
C) III only 
D) I and II only 
E) I and III only
F)I, II, and IV only
G)II and V only
H)I,II, III only

Question 6

Make a rough sketch of the graph of y = (1.1) $- x$ . Do not use a calculator.

Accepted Answer

The answer of Make a rough sketch of the graph...

Question 7

Relative to the graph of y = sin x, where x is in the radians, the graph of y = sin x, where x is in degrees, is changed in what way?

Accepted Answer

A) Compressed horizontally by a factor of $\frac { 180 } { \pi }$ 
B) Stretched vertically by a factor of $\frac { 180 } { \pi }$ 
C) Compressed horizontally by a factor of $\frac { 90 } { \pi }$ 
D) Stretched horizontally by a factor of $\frac { 90 } { \pi }$ 
E) Compressed vertically by a factor of $\frac { 90 } { \pi }$ 
F)Stretched vertically by a factor of $\frac { 90 } { \pi }$ 
G)Stretched horizontally by a factor of $\frac { 180 } { \pi }$ 
H)Compressed vertically by a factor of $\frac { 180 } { \pi }$ 
A) Compressed horizontally by a factor of $\frac { 180 } { \pi }$ 
B) Stretched vertically by a factor of $\frac { 180 } { \pi }$ 
C) Compressed horizontally by a factor of $\frac { 90 } { \pi }$ 
D) Stretched horizontally by a factor of $\frac { 90 } { \pi }$ 
E) Compressed vertically by a factor of $\frac { 90 } { \pi }$ 
F)Stretched vertically by a factor of $\frac { 90 } { \pi }$ 
G)Stretched horizontally by a factor of $\frac { 180 } { \pi }$ 
H)Compressed vertically by a factor of $\frac { 180 } { \pi }$

Question 8

Make a rough sketch of the graph of y = 3 (4 - 2 $x$ ). Do not use a calculator.

Accepted Answer

The answer of Make a rough sketch of the graph...

Question 9

Solve the equation $\log _ { 9 }$ (ln $x ^ { 3 }$ ) = 1

Accepted Answer

A)  $x = 3 ^ { e }$ 
B)  $x = 3 e$ 
C)  $x = e / 3$ 
D)  $x = 1$ 
E)  $x = e ^ { 2 }$ 
F) $x = 1 / e$ 
G) $x = 3 / e$ 
H) $x = e ^ { 3 }$ 
A)  $x = 3 ^ { e }$ 
B)  $x = 3 e$ 
C)  $x = e / 3$ 
D)  $x = 1$ 
E)  $x = e ^ { 2 }$ 
F) $x = 1 / e$ 
G) $x = 3 / e$ 
H) $x = e ^ { 3 }$

Question 10

Let f (x) = $\sqrt { 16 - x ^ { 2 } }$ . Find each of the following:
(a) f (0) $+$ f ( $- 2$ )
(b) f (x $+$ 2)
(c) [f (x)] $2$ (d) f (x $2$ )

Accepted Answer

(a) f (0) @#IMG-DLM& f ( @#IMG-DLM& ) = @#IMG-DLM& @#IMG-DLM& @#IMG-DLM& = 4 @#IMG-DLM& 2

Question 11

Find the smallest value in the domain of the function f (x) = $\sqrt { 2 x - 5 }$ .

Accepted Answer

A) 2 
B)  $\frac { 5 } { 2 }$ 
C) 5 
D)  $\frac { 2 } { 5 }$ 
E) -2
F)1
G)0

H)-5 
A) 2 
B)  $\frac { 5 } { 2 }$ 
C) 5 
D)  $\frac { 2 } { 5 }$ 
E) -2
F)1
G)0

H)-5

Question 12

Consider the data below:
$$\begin{array} { | c | c | c | c | c | c | c | } 
\hline t & 1 & 2 & 3 & 4 & 5 & 6 \
\hline y & 2.4 & 19 & 64 & 152 & 295 & 510 \
\hline
\end{array}$$
(a) Fit both an exponential curve and a third-degree polynomial to the data.(b) Which of the models appears to be a better fit? Defend your choice.

Accepted Answer

(a) @#IMG-DLM& (b) A third deg

Question 13

Find the range of the function $f ( x ) = \left\{ \begin{array} { l l } 
x ^ { 2 } - 4 x & \text { if } x \leq 2 \
| x - 4 | & \text { if } x > 2
\end{array} \right.$ .

Accepted Answer

A) [0,$\infty$). 
B) (-$\infty$,2] 
C) [-4, $\infty$). 
D) (-$\infty$, 0] 
E) [4, $\infty$)
F)(-$\infty$,4]
G)[2, $\infty$)
H)(-$\infty$,4] . 
A) [0,$\infty$). 
B) (-$\infty$,2] 
C) [-4, $\infty$). 
D) (-$\infty$, 0] 
E) [4, $\infty$)
F)(-$\infty$,4]
G)[2, $\infty$)
H)(-$\infty$,4] .

Question 14

A tank used for portland cement consists of a cylinder mounted on top of a cone, with its vertex pointing downward. The cylinder is 30 feet high, both the cylinder and the cone have radii of 4 feet, and the cone is 6 feet high.   (a) Determine the volume of cement contained in the tank as a function of the depth d of the cement.(b) What is the domain of the function?

Accepted Answer

(a) V (d)

Question 15

Decide whether the given function has an inverse. Defend your decision.

(a)  P (x) is the cost, in cents, of mailing a letter that weighs x ounces.

(b)  C (t) is the total number of cars which have driven past a particular point along a highway during a specific day where t represents the time, in hours, since midnight.

(c)  F (d) is the amount of fuel, in gallons, in your car when you have traveled d miles on a particular tankful of gas.

(d)  S (t) is the number of shoppers entering the Mall of America where t is the number of minutes past midnight on one particular day.

(e)  T (t) is the temperature inside an oven where t is the time, in minutes, from the beginning to the end of the oven's self-cleaning cycle.

(f)  V (x) is the volume of a cube whose side has length x.

(g)  f (n) is the number of students enrolled in your calculus class on the nth day of the term.

Accepted Answer

(a) P (x) does not have an inverse. The

Question 16

Find the range of the inverse for f (x) = $- \frac { 3 } { 5 + 2 x }$ .

Accepted Answer

A)  $\left( - \infty , - \frac { 5 } { 2 } \right)$ 
B) ( $- \infty$ , 0) 
C)  $\left( - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right)$ 
D)  $\left( - \infty , \frac { 5 } { 2 } \right)$ $\cup$ $\left( \frac { 5 } { 2 } \infty \right)$ 
E)  $\left( - \frac { 5 } { 2 } , \infty \right)$ 
F)(0, $\infty$ )

G) $\left( \frac { 5 } { 2 } , \infty \right)$ 
H) $\left( - \infty , - \frac { 5 } { 2 } \right)$ $\cup$ $\left( - \frac { 5 } { 2 } \infty \right)$ 
A)  $\left( - \infty , - \frac { 5 } { 2 } \right)$ 
B) ( $- \infty$ , 0) 
C)  $\left( - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right)$ 
D)  $\left( - \infty , \frac { 5 } { 2 } \right)$ $\cup$ $\left( \frac { 5 } { 2 } \infty \right)$ 
E)  $\left( - \frac { 5 } { 2 } , \infty \right)$ 
F)(0, $\infty$ )

G) $\left( \frac { 5 } { 2 } , \infty \right)$ 
H) $\left( - \infty , - \frac { 5 } { 2 } \right)$ $\cup$ $\left( - \frac { 5 } { 2 } \infty \right)$

Question 17

The position after t seconds of a projectile red with initial velocity v0 (measured in ft/s) at an angle above the horizontal from an initial height of h0 (measured in ft) is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t$ $y = \left( v _ { 0 } \sin \alpha \right) t - 16 t ^ { 2 } + h _ { 0 }$ .
(a) Eliminate the parameter t to show that the trajectory of the projectile is a parabola.
(b) If the projectile is red with an initial velocity of 300 ft/s from a height of 5 ft, what should be the angle of elevation in order to hit a target at the same height as the initial height, 900 ft downrange?
(c) If the projectile is red with an angle of elevation of 40S1U1P1$\circ$S1S1P0 and it strikes a target at the same height as the initial height, 600 ft downrange, what was the initial velocity of the projectile?

Accepted Answer

(a) @#LAT-DLM& , which is an equation of a parabol

Question 18

Find the value of log $2$ $\frac { 1 } { 8 }$ .

Accepted Answer

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

Question 19

Find the inverse function for f (x) = $\frac { x - 1 } { x + 1 }$ .

Accepted Answer

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

Question 20

A function has a domain [-4, 4] and a portion of its graph is shown.   (a) Complete the graph of f of its is known that f is an even function.(b) Complete the graph of f if it is known that f is an odd function.

Accepted Answer

The answer of A function has a domain [-4, 4]...

Determine the number of solutions of the equation 5x $4$ - 3x $2$ = 10x $5$ - 20x $6$ $+$ 63x $+$ 5.

Find the domain of the inverse for f (x) = $\sqrt { 2 x - 5 }$ .

Find functions f and g such that F(x) = 1 $\sqrt { 1 - \cos ^ { 2 } x }$ = $( f \circ g ) ( x )$

Which of the following are graphs of functions?

Make a rough sketch of the graph of y = (1.1) $- x$ . Do not use a calculator.

Relative to the graph of y = sin x, where x is in the radians, the graph of y = sin x, where x is in degrees, is changed in what way?

Make a rough sketch of the graph of y = 3 (4 - 2 $x$ ). Do not use a calculator.

Solve the equation $\log _ { 9 }$ (ln $x ^ { 3 }$ ) = 1

Let f (x) = $\sqrt { 16 - x ^ { 2 } }$ . Find each of the following: (a) f (0) $+$ f ( $- 2$ ) (b) f (x $+$ 2) (c) [f (x)] $2$ (d) f (x $2$ )

Find the smallest value in the domain of the function f (x) = $\sqrt { 2 x - 5 }$ .

Consider the data below: t 1 2 3 4 5 6 y 2.4 19 64 152 295 510 (a) Fit both an exponential curve and a third-degree polynomial to the data.(b) Which of the models appears to be a better fit? Defend your choice.

Find the range of the function $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } - 4 x & \text { if } x \leq 2 \\| x - 4 | & \text { if } x > 2\end{array} \right.$ .

Find the range of the inverse for f (x) = $- \frac { 3 } { 5 + 2 x }$ .

Find the value of log $2$ $\frac { 1 } { 8 }$ .

Find the inverse function for f (x) = $\frac { x - 1 } { x + 1 }$ .

A function has a domain [-4, 4] and a portion of its graph is shown. (a) Complete the graph of f of its is known that f is an even function.(b) Complete the graph of f if it is known that f is an odd function.

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Exam 1: Functions and Models

Determine the number of solutions of the equation 5x 444 - 3x 222 = 10x 555 - 20x 666 +++ 63x +++ 5.

Find the domain of the inverse for f (x) = 2x−5\sqrt { 2 x - 5 }2x−5​ .

Find functions f and g such that F(x) = 1 1−cos⁡2x\sqrt { 1 - \cos ^ { 2 } x }1−cos2x​ = (f∘g)(x)( f \circ g ) ( x )(f∘g)(x)

Which of the following are graphs of functions?

Make a rough sketch of the graph of y = (1.1) −x- x−x . Do not use a calculator.

Relative to the graph of y = sin x, where x is in the radians, the graph of y = sin x, where x is in degrees, is changed in what way?

Make a rough sketch of the graph of y = 3 (4 - 2 xxx ). Do not use a calculator.

Solve the equation log⁡9\log _ { 9 }log9​ (ln x3x ^ { 3 }x3 ) = 1

Let f (x) = 16−x2\sqrt { 16 - x ^ { 2 } }16−x2​ . Find each of the following: (a) f (0) +++ f ( −2- 2−2 ) (b) f (x +++ 2) (c) [f (x)] 222 (d) f (x 222 )

Find the smallest value in the domain of the function f (x) = 2x−5\sqrt { 2 x - 5 }2x−5​ .

Consider the data below: t 1 2 3 4 5 6 y 2.4 19 64 152 295 510 (a) Fit both an exponential curve and a third-degree polynomial to the data.(b) Which of the models appears to be a better fit? Defend your choice.

Find the range of the function f(x)={x2−4x if x≤2∣x−4∣ if x>2f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } - 4 x & \text { if } x \leq 2 \\| x - 4 | & \text { if } x > 2\end{array} \right.f(x)={x2−4x∣x−4∣​ if x≤2 if x>2​ .

Find the range of the inverse for f (x) = −35+2x- \frac { 3 } { 5 + 2 x }−5+2x3​ .

Find the value of log 222 18\frac { 1 } { 8 }81​ .

Find the inverse function for f (x) = x−1x+1\frac { x - 1 } { x + 1 }x+1x−1​ .

A function has a domain [-4, 4] and a portion of its graph is shown. (a) Complete the graph of f of its is known that f is an even function.(b) Complete the graph of f if it is known that f is an odd function.

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

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Vector Calculus

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Determine the number of solutions of the equation 5x $4$ - 3x $2$ = 10x $5$ - 20x $6$ $+$ 63x $+$ 5.

Find the domain of the inverse for f (x) = $\sqrt { 2 x - 5 }$ .

Find functions f and g such that F(x) = 1 $\sqrt { 1 - \cos ^ { 2 } x }$ = $( f \circ g ) ( x )$

Make a rough sketch of the graph of y = (1.1) $- x$ . Do not use a calculator.

Make a rough sketch of the graph of y = 3 (4 - 2 $x$ ). Do not use a calculator.

Solve the equation $\log _ { 9 }$ (ln $x ^ { 3 }$ ) = 1

Let f (x) = $\sqrt { 16 - x ^ { 2 } }$ . Find each of the following: (a) f (0) $+$ f ( $- 2$ ) (b) f (x $+$ 2) (c) [f (x)] $2$ (d) f (x $2$ )

Find the smallest value in the domain of the function f (x) = $\sqrt { 2 x - 5 }$ .

Find the range of the function $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } - 4 x & \text { if } x \leq 2 \\| x - 4 | & \text { if } x > 2\end{array} \right.$ .

Find the range of the inverse for f (x) = $- \frac { 3 } { 5 + 2 x }$ .

Find the value of log $2$ $\frac { 1 } { 8 }$ .

Find the inverse function for f (x) = $\frac { x - 1 } { x + 1 }$ .