Practice Test 9 - AP Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

Based on the logarithmic expression provided, simplify $\log_7\!\left(\tfrac{49x}{y^2}\right)-\log_7(x)+2\log_7(y)$ for $x,y>0$ .
A decoding step needs $A^{-1}$ for $A=\begin{bmatrix}5&1\\2&1\end{bmatrix}$ . Using $A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ , calculate the inverse, if possible.
A population model is $N(t)=0.3t^3-2t^2+4t+10$ ; the rate is $N'(t)$ . What is the instantaneous rate of change at $t=5$ ?
A rational function is a quotient of polynomials, $f(x)=\frac{p(x)}{q(x)}$ . Zeros are values of $x$ that make $f(x)=0$ , so the numerator must be zero while the denominator is not. Vertical asymptotes occur where the denominator equals zero and does not cancel with the numerator. Consider $f(x)=\frac{x^2+1}{x^2-1}.$ Since $x^2-1=0$ at $x=\pm 1$ , the function is undefined there and has vertical asymptotes at those $x$ -values. The numerator $x^2+1$ has no real roots, so there are no real zeros. Using the function provided, identify the $x$ -values for which $f(x)=\frac{x^2+1}{x^2-1}$ is undefined.
According to the model, approximately how many subscribers, in millions, did the service have at the beginning of 2018?
Consider $m(x)=\dfrac{x^2-2x+1}{x-1}$ . Which value of $x$ makes numerator and denominator zero, creating a hole?
Based on the situation described, what does the rational population model imply about population as $t\to\infty$ ?
A savings account has $\$2{,}500$ invested at $4.8\%$ annual interest, compounded monthly, and it grows to $\$4{,}000$ . You write $4000=2500\left(1+\frac{0.048}{12}\right)^{12t}$ and plan to solve for $t$ using logarithms and change of base. Based on the scenario above, determine the time $t$ in years required for the investment to reach $\$4{,}000$ . Use $\log_b(x)=\frac{\ln(x)}{\ln(b)}$ if needed. Round to the nearest tenth.
In $\mathbf{r}(t)=\langle (v\cos a)t,\,(v\sin a)t-4.9t^2\rangle$ , which term causes the downward curvature over time?
Based on the problem described, determine the parametrization for $\frac{x^2}{16}+\frac{y^2}{9}=1$ using $t$ .
A school tracks users of a new homework app each week. Model 1: Exponential, $U(t)=40(1.30)^t$ , uses base $1.30$ for rapid word-of-mouth growth. Model 2: Logarithmic, $U(t)=35+120\log(0.4t+1)$ , uses coefficient $120$ to model fast early adoption that slows as most students join. After week 8, nearly everyone who wants the app already has it, and weekly increases shrink. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?
The average rates of change of a polynomial function $f(x)$ over consecutive equal-length intervals can be modeled by a linear function with a positive slope. Which of the following statements about $f(x)$ must be true?
A function $N(p)$ is created to model the number of tickets sold for a concert as a function of the ticket price $p$ in dollars. Which of the following describes a necessary restriction on the range of this function model?
### Semi-Log Scales A semi-log plot uses a logarithmic scale on one axis, usually the $y$ -axis, and a linear scale on the other. This is helpful when $y$ values span several powers of 10. ### Why Use Semi-Log Plots? Semi-log plots can turn an exponential model into a straight line. - If $N(t)=N_0(1+r)^t$ , then $\log(N)$ changes linearly with $t$ . - This makes it easier to recognize constant-percent growth. ### Scenario: Population Growth A city's population $N$ is plotted against time $t$ on a semi-log graph (log $y$ -axis). The points form a straight line. ### Interpreting a Straight Line On a semi-log plot with log $y$ -axis: - a straight line indicates exponential growth, - and the slope reflects the constant percent growth rate. In the example of population growth, what does a straight line on a semi-log plot indicate about the data presented?
A Ferris wheel height is $h(t)=10\cos\left(\frac{\pi}{12}t+\frac{\pi}{3}\right)+12$ , with $t$ in minutes. The amplitude is the wheel radius, and the midline is the axle height. The phase shift indicates when the rider reaches the top relative to $t=0$ . Using the given function, determine the phase shift of the function and explain its significance.
Based on the discussion of inverse functions, what is the domain of $f^{-1}$ if $f(x)=2^x$ is restricted to $x\ge 1$ ?
Based on the discussion of inverse functions, which function is the inverse of $f(x)=e^{x}-7$ ?
Let $A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}$ . What is the inverse matrix, $A^{-1}$ ?
A student invests money in a certificate of deposit (CD). The initial deposit is $P_0=\$1{,}800$ , and the CD earns $3.6\%$ annual interest compounded quarterly. The balance after $t$ years is modeled by $A(t)=P_0\left(1+\frac{0.036}{4}\right)^{4t}.$ A table of modeled balances (rounded to the nearest dollar) is shown below. $t$ (years): 0, 2, 4, 6 $A(t)$ (USD): 1800, 1934, 2078, 2233 In this context, $P_0$ is the starting amount deposited, and the factor $\left(1+\frac{0.036}{4}\right)$ represents the growth each quarter. The exponent $4t$ counts the number of compounding periods, so increasing $t$ increases the number of times the balance is multiplied by the quarterly growth factor. Students compare the table to the formula to confirm that the dollar increase is not constant; instead, the balance grows by a constant percentage per year. They also discuss how changing the interest rate or compounding frequency would change the long-term balance. Based on the scenario above, what does the initial value $P_0$ represent in this context?
A car drives in a circle of radius $50\text{ m}$ centered at the origin: $x(t)=50\cos(0.2t)$ , $y(t)=50\sin(0.2t)$ . What is its speed at $t=7$ ?

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