Question 1

Find the value of $\log _ { 16 } 8$ .
A) $\frac { 1 } { 4 }$ 
B) $\frac { 1 } { 2 }$ 
C) $\frac { 3 } { 4 }$ 
D)1
E) $\frac { 3 } { 2 }$ 
F)2
G)3
H)4

Accepted Answer

The value of $\log _ { 16 } 8$ can be simplified by converting both numbers to a common base. Both 16 and 8 are powers of 2, with 16 being $2^4$ and 8 being $2^3$. Thus, $\log _ { 16 } 8 = \log _ { 2^4 } 2^3$. Using the logarithm power rule, this simplifies to $\frac { 3 } { 4 }$.

Question 2

(a) Sketch the curves represented by:
(i) x = t/2, y = $1 - \frac { t ^ { 2 } } { 4 }$ (ii) x = sin t, y = 1 sin $2$ t
(iii) x = e $t$ , y = 1 - e $2 t$ (b) Describe the differences between the three curves in part (a).(c) Produce Cartesian equations for the curves in parts (a)(i), (ii), and (iii) by eliminating the parameter. Compare your results.

Accepted Answer

(b) (i) This curve represents the entire parabola y = 1 $- x ^ { 2 }$ .
(ii) This curve represents the part of the parabola in (i) with x $\in$ [ - 1, 1].
(iii) This curve represents the part of the parabola in (i) with 0 ≤ x.
(c) They have the same Cartesian equation, y = 1 - x $2$ with different domains.

Question 3

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

Accepted Answer

To find the inverse function, swap $x$ and $y$ and solve for $y$. Starting with $y = \frac{x - 1}{x + 1}$, swapping gives $x = \frac{y - 1}{y + 1}$. Solving for $y$, we get $y = \frac{1 + x}{1 - x}$, which matches choice D.

Question 4

Find the domain of the inverse for f (x) = $\sqrt { 2 x - 5 }$ .
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) $\left[ \frac { 5 } { 2 } , \infty \right)$

Accepted Answer

The domain of the inverse function is the range of the original function. For $ f(x) = \sqrt{2x - 5} $, the expression under the square root must be non-negative, so $ 2x - 5 \geq 0 $, which gives $ x \geq \frac{5}{2} $. Therefore, the range of $ f(x) $ is $[0, \infty)$, which is the domain of the inverse function.

Question 5

Eliminate the parameter in the equations $x = t ^ { 2 } , y = t ^ { 4 }$ 
A) $y = x ^ { 2 } \text { for } x \geq 0$ 
B) $y = \sqrt { x } \text { for } x \geq 0$ 
C) $y = 2 x ^ { 2 } \text { for } x \geq 0$ 
D) $y = \sqrt { 2 x } \text { for } x \geq 0$ 
E) $y = 2 \sqrt { x } \text { for } x \geq 0$ 
F) $y = x ^ { 2 } / 2 \text { for } x \geq 0$ 
G) $y = \sqrt { x } / 2 \text { for } x \geq 0$ 
H) $y = \sqrt { x / 2 } \text { for } x \geq 0$

Accepted Answer

Given $x = t^2$, solving for $t$ gives $t = \sqrt{x}$ (considering $t \geq 0$ since $x \geq 0$). Substituting $t = \sqrt{x}$ into $y = t^4$ gives $y = (\sqrt{x})^4 = x^2$, hence $y = x^2$ for $x \geq 0$.

Question 6

A baseball slugger hits a knee-high pitch toward the outfield. Suppose that the position of the baseball after t seconds is given by :
x = (v $o$ cos $\alpha$ ) t, y = (v $o$ sin $\alpha$ ) t $- 16 t ^ { 2 } + h _ { o }$ where v $o$ is the velocity in feet per second at which the ball leaves the bat at an angle $\alpha$ to the horizontal and from a height h $o$ above the ground.(a) Suppose that the ball is struck 2 feet above the ground with an initial velocity of 120 ft/sec and at an angle of 35 degrees.(i) When will the ball strike the ground?
(ii) How far will the ball travel (horizontally) before it touches the ground?
(iii) What is the maximum height reached by the ball?
(iv) Will the ball clear the 10 ft tall outfield fence, which is 410 feet away from the point where the ball is struck?
(b) If the ball is struck 2 feet above the ground at an initial velocity of 120 ft/sec and at an angle of 55 degrees:
(i) When will the ball strike the ground?
(ii) How far will the ball travel (horizontally) before it touches the ground?
(iii) What is the maximum height reached by the ball?
(iv) Will the ball clear the 10' tall outfield fence, which is 410 feet away from the point where the ball is struck?

Accepted Answer

The answer of A baseball slugger hits a knee-high pitch...

Question 7

Consider the pairs of parametric equations $x = 2 - 3 t , y = 7 - 6 t$ and $x = t - 1 , y = 1 + 2 t$ (a) Show that these pairs of equations produce the same line.(b) What are the slope and y-intercept of this line?

Accepted Answer

The answer of Consider the pairs of parametric equations \(x...

Question 8

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

Accepted Answer

The answer of Find the range of the inverse for...

Question 9

Find the value of log $2$ $\frac { 1 } { 8 }$ .
A) $\frac { 1 } { 4 }$ 
B) $\frac { 1 } { 3 }$ 
C)0
D)1
E) - 1
F)2
G) - 2
H) - 3

Accepted Answer

The expression log base 2 of 1/8 can be rewritten as log base 2 of 2^(-3), which equals -3.

Question 10

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

The answer of The position after t seconds of a...

Question 11

Sketch a possible parametric curve defined by the following table of values. $$\begin{array} { | l | c c c c c c c c c | } 
\hline t & 0 & \frac { \pi } { 4 } & \frac { \pi } { 2 } & \frac { 3 \pi } { 4 } &{ \pi } & \frac { 5 \pi } { 4 } & \frac { 3 \pi } { 2 } & \frac { 7 \pi } { 4 } & 2 \pi \
\hline x & 2 & \sqrt { 2 } & 0 & - \sqrt { 2 } & - 2 & - \sqrt { 2 } & 0 & \sqrt { 2 } & 2 \
\hline y & 0 & \sqrt { 2 } & - 2 & \sqrt { 2 } & 0 & - \sqrt { 2 } & 2 & - \sqrt { 2 } & 0 \
\hline
\end{array}$$

Accepted Answer

The answer of Sketch a possible parametric curve defined by...

Question 12

Given the function tan x with domain $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ find the domain of its inverse.
A) $\left[ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right]$ 
B)[0, $\infty$ )
C)[ $- \pi , \pi$ ]
D)[ $- 1,1$ ]
E) $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ 
F) $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ 
G)( $- \infty , \infty )$ 
H) $\left[ - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right]$

Accepted Answer

The inverse of the tangent function, $\tan^{-1}(x)$ or arctan(x), has a domain of all real numbers, $(-\infty, \infty)$, because tangent values can range from negative infinity to positive infinity within its original domain of $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Question 13

Suppose that the position of one particle at time t is given by
x $1$ = 4 sin t, y $1$ = 2 cos t for 0 $\leq t \leq 2 \pi$ and the position of a second particle is given by
x $2$ = 2 cos t - 1, y $2$ = 4 sin t + 2 for 0 $\leq t \leq 2 \pi$ (a) How many points of intersection are there for these two paths?
(b) Are any of these collision points (that is, points where the particles are at the same place at the same time)? If so, find the collision points.

Accepted Answer

The answer of Suppose that the position of one particle...

Question 14

At how many places does the curve x = cos 3t, y = sin t cross the x-axis?
A)1
B)2
C)3
D)4
E)5
F)6
G)7
H)8

Accepted Answer

To find the points where the curve crosses the x-axis, we need to find the values of t when y=0. From the given equation, we have y=sin(t), so sin(t)=0 when t=0, π, 2π, … (kπ, where k is an integer). 
Substituting these values of t into x=cos(3t), we get x=cos(0)=1, cos(3π)=-1, cos(6π)=1, ... 
So the curve crosses the x-axis at x=1 and x=-1. Therefore, the number of places where the curve crosses the x-axis is 2, which is option (B).

Question 15

Eliminate the parameter in the equations $2 = \tan t , y = 2 \cos ^ { 2 } t , 0 \leq t \leq 2 \pi$ 
A) $y = 2 x ^ { 2 }$ 
B) $y = x ^ { 2 } + 4$ 
C) $y = 8 x ^ { 2 } + 32$ 
D) $y = \frac { x ^ { 2 } + 4 } { 8 }$ 
E) $y = x ^ { 2 } + 32$ 
F) $y = 4 x ^ { 2 } + 8$ 
G) $y = 8 x ^ { 2 } + 4$ 
H) $y = \frac { 8 } { x ^ { 2 } + 4 }$

Accepted Answer

The answer of Eliminate the parameter in the equations \(2...

Question 16

Describe the curve defined by x = cos t, y = sin $2$ t.
A)Circle
B)Semicircle
C)Quarter-circle
D)Parabola
E)Portion of a parabola
F)Hyperbola
G)Single branch of a hyperbola
H)Portion of branch of hyperbola

Accepted Answer

The answer of Describe the curve defined by x =...

Question 17

Describe the motion of a particle with position x = 2 + cos t, y = 3 + sin t as t varies in the interval [0, 2 $\pi$ ].

Accepted Answer

The answer of Describe the motion of a particle with...

Question 18

Find the inverse function for f (x) = $\frac { 5 - 2 x } { 3 }$ .
A) $\frac { 5 - 3 x } { 2 }$ 
B) $\frac { 2 x - 5 } { 3 }$ 
C) $\frac { 2 - 3 x } { 5 }$ 
D) $\frac { 3 x - 2 } { 5 }$ 
E) $\frac { 2 x + 5 } { 3 }$ 
F) $\frac { 3 } { 5 - 2 x }$ 
G) $\frac { 2 } { 5 - 3 x }$ 
H) $3 x - 5$

Accepted Answer

To find the inverse function, swap $x$ and $y$ and solve for $y$. Starting with $y = \frac{5 - 2x}{3}$, swapping gives $x = \frac{5 - 2y}{3}$. Solving for $y$ gives $y = \frac{5 - 3x}{2}$, which matches choice A.

Question 19

Sketch a possible parametric curve defined by the following table of values. $$\begin{array} { | l | c c c c c c c c c | } 
\hline t & 0 & \frac { \pi } { 4 } & \frac { \pi } { 2 } & \frac { 3 \pi } { 4 } & \pi & \frac { 5 \pi } { 4 } & \frac { 3 \pi } { 2 } & \frac { 7 \pi } { 4 } & 2 \pi \
\hline x & 0 & 0.18 & 0 & - 0.53 & - 1 & - 0.88 & 0 & 1.24 & 2 \
\hline y & 0 & 0.18 & 0.5 & 0.53 & 0 & - 0.88 & - 1.5 & - 1.24 & 0 \
\hline
\end{array}$$

Accepted Answer

The answer of Sketch a possible parametric curve defined by...

Question 20

Find the value of log $1 / 2$ 1.
A) - 1
B) - $\frac { 1 } { 2 }$ 
C)0
D)10 $2$ 
E)1
F) $\frac { 1 } { 2 }$ 
G)2

H) - 2

Accepted Answer

The logarithm of 1 to any base is always 0.

Quiz 1: Functions and Models

Find the value of log⁡168\log _ { 16 } 8log16​8 .

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

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

Eliminate the parameter in the equations x=t2,y=t4x = t ^ { 2 } , y = t ^ { 4 }x=t2,y=t4

Consider the pairs of parametric equations x=2−3t,y=7−6tx = 2 - 3 t , y = 7 - 6 tx=2−3t,y=7−6t and x=t−1,y=1+2tx = t - 1 , y = 1 + 2 tx=t−1,y=1+2t (a) Show that these pairs of equations produce the same line.(b) What are the slope and y-intercept of this line?

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​ .

Given the function tan x with domain (−π2,π2) \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) (−2π​,2π​) find the domain of its inverse.

At how many places does the curve x = cos 3t, y = sin t cross the x-axis?

Eliminate the parameter in the equations 2=tan⁡t,y=2cos⁡2t,0≤t≤2π2 = \tan t , y = 2 \cos ^ { 2 } t , 0 \leq t \leq 2 \pi2=tant,y=2cos2t,0≤t≤2π

Describe the curve defined by x = cos t, y = sin 222 t.

Describe the motion of a particle with position x = 2 + cos t, y = 3 + sin t as t varies in the interval [0, 2 π\piπ ].

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

Find the value of log 1/21 / 21/2 1.

Find the value of $\log _ { 16 } 8$ .

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

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

Eliminate the parameter in the equations $x = t ^ { 2 } , y = t ^ { 4 }$

Consider the pairs of parametric equations $x = 2 - 3 t , y = 7 - 6 t$ and $x = t - 1 , y = 1 + 2 t$ (a) Show that these pairs of equations produce the same line.(b) What are the slope and y-intercept of this line?

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

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

Given the function tan x with domain $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ find the domain of its inverse.

Eliminate the parameter in the equations $2 = \tan t , y = 2 \cos ^ { 2 } t , 0 \leq t \leq 2 \pi$

Describe the curve defined by x = cos t, y = sin $2$ t.

Describe the motion of a particle with position x = 2 + cos t, y = 3 + sin t as t varies in the interval [0, 2 $\pi$ ].

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

Find the value of log $1 / 2$ 1.