Question 1

For which value of $x$ does $4 x ^ { 2 } - 10 x + 15$ reach a minimum? Restrict your attention to the horizontal span of 0 to 10.

Accepted Answer

The minimum value of a quadratic function $ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. For the given function $4x^2 - 10x + 15$, $a = 4$ and $b = -10$, so the minimum occurs at $x = -\frac{-10}{2*4} = \frac{10}{8} = 1.25$.

Question 2

Solve the equation Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

The answer of Solve the equation Restrict your attention to...

Question 3

Solve the inequality $7 < x ^ { 3 } - 1 < 26$ . Restrict your attention to positive values of $x$ .

Accepted Answer

To solve the inequality $7 < x^3 - 1 < 26$, we first add 1 to all parts of the inequality to get $8 < x^3 < 27$. Taking the cube root of all parts gives $2 < x < 3$, since the cube root of 8 is 2 and the cube root of 27 is 3. This solution is restricted to positive values of $x$, which fits the given range.

Question 4

Which of the following is the graph of $\frac { x ^ { 2 } + 24 } { x + 8 } 0$ on a horizontal span of 0 to 4.5?

Accepted Answer

To determine the graph of the function, we can simplify the expression first:

$$\frac { x ^ { 2 } + 24 } { x + 8 }=\frac {(x+8)(x-8)+88}{x+8}=x-8+\frac{88}{x+8}$$

We can see that as x approaches infinity or negative infinity, the curve approaches the line y=x-8. Also, there is a vertical asymptote at x=-8 (where the denominator is zero). 
Option C matches these characteristics with the correct intercepts and asymptotes, while the other options do not match the characteristics of the function.

Question 5

Find the maximum value of $11 + 9 x - 7 x ^ { 2 }$ Restrict your attention to the horizontal span of 0 to 9.

Accepted Answer

We are given an equation in terms of x and asked to find the maximum value of the equation on the interval from 0 to 9. To do this, we will take the derivative of the equation and set it equal to 0 to find the critical point(s) where the maximum or minimum occurs.

$$f(x) = 11 + 9x - 7x^2$$

Taking the derivative:

$$f'(x) = 9 - 14x$$

Setting equal to 0:

$$9 - 14x = 0$$
$$x = \frac{9}{14}$$

This critical point occurs within the given interval, so we will check the value of the function at that point and at the endpoints of the interval (0 and 9).

$$f(0) = 11 + 0 - 7(0)^2 = 11$$
$$f(9) = 11 + 9(9) - 7(9)^2 = -352$$
$$f(9/14) = 11 + 9\left(\frac{9}{14}\right) - 7\left(\frac{9}{14}\right)^2 \approx 13.89$$

Therefore, the maximum value of the function on the interval from 0 to 9 occurs at x = 9/14, and is approximately 13.89. The correct answer is B.

Question 6

Find the maximum value of $x ^ { 4 } e ^ { - 19 x }$ . Restrict your attention to the horizontal span of 0 to

Accepted Answer

To find the maximum value of the function $f(x) = x^4e^{-19x}$, we can use calculus. Taking the derivative of $f(x)$ with respect to $x$ and setting it to zero will give us the critical points. The derivative is $f'(x) = 4x^3e^{-19x} - 19x^4e^{-19x}$. Setting $f'(x) = 0$ and solving for $x$ gives $x = \frac{4}{19}$, which is approximately 0.21. Evaluating $f(x)$ at $x = \frac{4}{19}$ will give us the maximum value since the second derivative test confirms it's a maximum. The exact maximum value isn't provided in the options, but the process of finding it involves calculus and the critical point calculation.

Question 7

You got a birthday gift of $42.62, which you choose to spend on music.You pay $12.49 for an iPad app that lets you download songs for $1.31 each. How many songs can you download before your
Birthday money is used up?

Accepted Answer

First, subtract the cost of the iPad app from your total amount of money: 
$42.62 - $12.49 = $30.13
Next, divide the remaining amount of money by the cost of each song: 
$30.13 ÷ $1.31 per song = 22.99 songs (rounded to the nearest whole number). 
Therefore, you can purchase 23 songs before running out of money.

Question 8

$$\text { Which of the following is the graph of } e ^ { x } - 4.3 x ^ { 2 } \text { on a horizontal span of } 0 \text { to } 4.5 ?$$ '

Accepted Answer

The answer of \[\text { Which of the following is...

Question 9

What is the minimum value of $6 x ^ { 3 } - 11 x + 16 ?$ Restrict your attention to the horizontal span of 0 to 11.

Accepted Answer

The answer of What is the minimum value of \(6...

Question 10

Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

Firstly, we notice that $x>0$ since the denominator $1+2^x$ is always positive. Plotting the graphs of $y=20/(1+2^x)$ and $y=x$, we can see that they intersect at one point in the interval $(0,5)$. To determine this intersection point, we can use numerical methods such as the bisection method or Newton-Raphson method. Using any of these methods, we can find that $x\approx2.69$. Thus, the answer is $\boxed{	extbf{(C)}}$.

Note: It is important to check that this is the only solution in the given interval by checking the concavity of the functions. In this case, we can show that $y=20/(1+2^x)$ is concave up while $y=x$ is concave down, which implies that they can intersect at most twice. Since we have found one intersection, we can conclude that there is only one solution in the given interval.

Question 11

Solve the inequality $6 < 3.1 \times 1.6 ^ { x } < 17 .$

Accepted Answer

To solve the inequality $6 < 3.1 \times 1.6^x < 17$, we first divide all parts of the inequality by 3.1, getting $6/3.1 < 1.6^x < 17/3.1$. Simplifying the fractions gives approximately $1.935 < 1.6^x < 5.484$. Taking the logarithm base 1.6 of all parts (since we're dealing with $1.6^x$) gives us $\log_{1.6}(1.935) < x < \log_{1.6}(5.484)$. Calculating these logarithms gives approximately $1.41 < x < 3.62$, which matches option A.

Question 12

Solve the equation $$16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 }$$ Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

Substitute each choice into the equation to check which one satisfies it. Only 2.06 satisfies the equation within the given range.

Question 13

For which value of $x$ does $6 + 11 x - 7 x ^ { 2 }$ reach a maximum? Restrict your attention to the horizontal span of 0 to 11.

Accepted Answer

The answer of For which value of $x$ does \(6...

Question 14

Solve the equation Restrict your attention to the horizontal span of 0 to 5. $3 e ^ { x } = 68$

Accepted Answer

The answer of Solve the equation Restrict your attention to...

Question 15

The amount $A$ , in grams, of a radioactive substance remaining after  years is given by $A = 14 \times 0.55 ^ {t}$ What is the limiting value for the amount of the radioactive substance?

Accepted Answer

As time approaches infinity, the exponent in the expression approaches negative infinity, making the whole expression approach zero. So the limiting value for the amount of the radioactive substance is 0 grams.

Question 16

The life expectancy E , in years, for a child born  years after 2000 is given by $E = 76.62 + 0.18 t$ In what birth year would a child have a life expectancy of 78.78 years?

Accepted Answer

Substitute 78.78 for E in the equation and solve for t:
$78.78 = 76.62 + 0.18t$
$2.16 = 0.18t$
$t = 12$
Therefore, a child born 12 years after 2000 would have a life expectancy of 78.78 years. 
Adding 12 years to 2000 gives 2012, so the answer is (B) 2012.

Question 17

Solve the inequality

Accepted Answer

The answer of Solve the inequality 
...

Question 18

Solve the inequality $8 - x ^ { 2 } > 5 + x ^ { 4 }$ Be careful. The graphs cross twice.

Accepted Answer

The answer of Solve the inequality \(8 - x ^...

Question 19

A rental car charges $61.97 for insurance and other fees plus a flat rate of $28.42 per day. Write a formula that gives the cost C , in dollars, of renting a car for  days.

Accepted Answer

The answer of A rental car charges $61.97 for insurance...

Quiz 2: Graphical and Tabular Analysi

For which value of $x$ does $4 x ^ { 2 } - 10 x + 15$ reach a minimum? Restrict your attention to the horizontal span of 0 to 10.

Solve the equation Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality $7 < x ^ { 3 } - 1 < 26$ . Restrict your attention to positive values of $x$ .

Which of the following is the graph of $\frac { x ^ { 2 } + 24 } { x + 8 } 0$ on a horizontal span of 0 to 4.5?

Find the maximum value of $11 + 9 x - 7 x ^ { 2 }$ Restrict your attention to the horizontal span of 0 to 9.

Find the maximum value of $x ^ { 4 } e ^ { - 19 x }$ . Restrict your attention to the horizontal span of 0 to

You got a birthday gift of $42.62, which you choose to spend on music.You pay $12.49 for an iPad app that lets you download songs for $1.31 each. How many songs can you download before your Birthday money is used up?

$\text { Which of the following is the graph of } e ^ { x } - 4.3 x ^ { 2 } \text { on a horizontal span of } 0 \text { to } 4.5 ?$ '

What is the minimum value of $6 x ^ { 3 } - 11 x + 16 ?$ Restrict your attention to the horizontal span of 0 to 11.

Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality $6 < 3.1 \times 1.6 ^ { x } < 17 .$

Solve the equation $16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 }$ Restrict your attention to the horizontal span of 0 to 5.

For which value of $x$ does $6 + 11 x - 7 x ^ { 2 }$ reach a maximum? Restrict your attention to the horizontal span of 0 to 11.

Solve the equation Restrict your attention to the horizontal span of 0 to 5. $3 e ^ { x } = 68$

The life expectancy E , in years, for a child born years after 2000 is given by $E = 76.62 + 0.18 t$ In what birth year would a child have a life expectancy of 78.78 years?

Solve the inequality

Solve the inequality $8 - x ^ { 2 } > 5 + x ^ { 4 }$ Be careful. The graphs cross twice.

A rental car charges $61.97 for insurance and other fees plus a flat rate of $28.42 per day. Write a formula that gives the cost C , in dollars, of renting a car for days.

Quiz 2: Graphical and Tabular Analysi

For which value of xxx does 4x2−10x+154 x ^ { 2 } - 10 x + 154x2−10x+15 reach a minimum? Restrict your attention to the horizontal span of 0 to 10.

Solve the equation Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality 7<x3−1<267 < x ^ { 3 } - 1 < 267<x3−1<26 . Restrict your attention to positive values of xxx .

Which of the following is the graph of x2+24x+80\frac { x ^ { 2 } + 24 } { x + 8 } 0x+8x2+24​0 on a horizontal span of 0 to 4.5?

Find the maximum value of 11+9x−7x211 + 9 x - 7 x ^ { 2 }11+9x−7x2 Restrict your attention to the horizontal span of 0 to 9.

Find the maximum value of x4e−19xx ^ { 4 } e ^ { - 19 x }x4e−19x . Restrict your attention to the horizontal span of 0 to

You got a birthday gift of $42.62, which you choose to spend on music.You pay $12.49 for an iPad app that lets you download songs for $1.31 each. How many songs can you download before your Birthday money is used up?

Which of the following is the graph of ex−4.3x2 on a horizontal span of 0 to 4.5?\text { Which of the following is the graph of } e ^ { x } - 4.3 x ^ { 2 } \text { on a horizontal span of } 0 \text { to } 4.5 ? Which of the following is the graph of ex−4.3x2 on a horizontal span of 0 to 4.5? '

What is the minimum value of 6x3−11x+16?6 x ^ { 3 } - 11 x + 16 ?6x3−11x+16? Restrict your attention to the horizontal span of 0 to 11.

Solve the equation 201+2x=x\frac { 20 } { 1 + 2 ^ { x } } = x1+2x20​=x Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality 6<3.1×1.6x<17.6 < 3.1 \times 1.6 ^ { x } < 17 .6<3.1×1.6x<17.

Solve the equation 16x2+4x+1=4x+500016 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 }16x2+4x+1​=4x+5000​ Restrict your attention to the horizontal span of 0 to 5.

For which value of xxx does 6+11x−7x26 + 11 x - 7 x ^ { 2 }6+11x−7x2 reach a maximum? Restrict your attention to the horizontal span of 0 to 11.

Solve the equation Restrict your attention to the horizontal span of 0 to 5. 3ex=683 e ^ { x } = 683ex=68

The amount AAA , in grams, of a radioactive substance remaining after years is given by A=14×0.55tA = 14 \times 0.55 ^ {t}A=14×0.55t What is the limiting value for the amount of the radioactive substance?

The life expectancy E , in years, for a child born years after 2000 is given by E=76.62+0.18tE = 76.62 + 0.18 tE=76.62+0.18t In what birth year would a child have a life expectancy of 78.78 years?

Solve the inequality

Solve the inequality 8−x2>5+x48 - x ^ { 2 } > 5 + x ^ { 4 }8−x2>5+x4 Be careful. The graphs cross twice.

A rental car charges $61.97 for insurance and other fees plus a flat rate of $28.42 per day. Write a formula that gives the cost C , in dollars, of renting a car for days.

For which value of $x$ does $4 x ^ { 2 } - 10 x + 15$ reach a minimum? Restrict your attention to the horizontal span of 0 to 10.

Solve the inequality $7 < x ^ { 3 } - 1 < 26$ . Restrict your attention to positive values of $x$ .

Which of the following is the graph of $\frac { x ^ { 2 } + 24 } { x + 8 } 0$ on a horizontal span of 0 to 4.5?

Find the maximum value of $11 + 9 x - 7 x ^ { 2 }$ Restrict your attention to the horizontal span of 0 to 9.

Find the maximum value of $x ^ { 4 } e ^ { - 19 x }$ . Restrict your attention to the horizontal span of 0 to

$\text { Which of the following is the graph of } e ^ { x } - 4.3 x ^ { 2 } \text { on a horizontal span of } 0 \text { to } 4.5 ?$ '

What is the minimum value of $6 x ^ { 3 } - 11 x + 16 ?$ Restrict your attention to the horizontal span of 0 to 11.

Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality $6 < 3.1 \times 1.6 ^ { x } < 17 .$

Solve the equation $16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 }$ Restrict your attention to the horizontal span of 0 to 5.

For which value of $x$ does $6 + 11 x - 7 x ^ { 2 }$ reach a maximum? Restrict your attention to the horizontal span of 0 to 11.

Solve the equation Restrict your attention to the horizontal span of 0 to 5. $3 e ^ { x } = 68$

The life expectancy E , in years, for a child born years after 2000 is given by $E = 76.62 + 0.18 t$ In what birth year would a child have a life expectancy of 78.78 years?

Solve the inequality $8 - x ^ { 2 } > 5 + x ^ { 4 }$ Be careful. The graphs cross twice.