AP Precalc MCQ Final
The binomial theorem can be used to expand the polynomial function p, given by p(x)=(x-3)^5. What is the coefficient of the x^3 term in the expanded polynomial?
(-3)^2 * 10
The location of point X in polar coordinates (r, theta) is (1, 5pi/6). Which of the following describes the location of point X in rectangular coordinates (x, y)?
(-3pi/2, 1/2)
The polar function r=f(theta), where f(theta)=1+2cos(theta), is graphed in the polar coordinate system for 0≤theta≤2pi. On which of the following intervals of theta is the distance between the point with polar coordinates (f(theta), theta) and the origin decreasing?
(0,2.094) and (3.142,4.189)
The function C models the cost, in dollars, for producing x items and is given by C(x)=1000+bx/x, where b is a constant. It is known that the cost is $115 to produce 10 items and $65 to produce 20 items. What is the average rate of change of C as x changes from x=30 to x=40 ?
-$0.83 per item
The table gives values for a polynomial function f at selected values of x. What is the average rate of change of f over the closed interval [1,4]?
-2
The figure shows a circle centered at the origin with an angle of measure theta radians in standard position. The terminal ray of the angle intersects the circle at point P. If tanx=-34, what is the slope of the line that passes through the origin and P?
-3/4
Consider the functions g and h given by g(x)=4^x and h(x)=16^x+2. In the xy-plane, what is the x-coordinate of the point of intersection of the graphs of g and h?
-4
The table shows the length of high-speed railways, in thousands of kilometers, in a certain country, starting in the year 2010 (t=0). A quartic regression is used to model the length of high-speed railways as a function of years since 2010. What length of high-speed railways, in thousands of kilometers, is predicted by the model for 2028 (t=18) ?
127.0
The function f is given by f(x)=log2(x). What input value in the domain of f yields an output value of 4?
16
At time t=0 years, the population of a certain city was 23,144. During each of the next 10 years, the population decreased by 4% per year. Based on this information, which of the following models the population as a function of time t, in years, for 0<t<10?
23,144(0.96)^t
The function f is given by f(x)=2cos(pix)+3. The graph of f is mapped to the graph of g in the same xy-plane by a horizontal translation of the graph of f by pi/2 units right. Which of the following is an expression for g(x)?
2cos(pi(x-pi/2))+3
The figure shows the graph of an exponential decay function f. The coordinates of two of the points are labeled. If y=f(x), what is the y-coordinate of the point on the graph where x=0?
40
The table gives the weight y, in pounds, of an animal for selected ages x, in years. A logarithmic regression is used to model these data. What is the weight of the animal, to the nearest pound, predicted by the logarithmic function model at age 4.5 years?
5345
The figure shows a circle centered at the origin with an angle of measure x radians in standard position and point P on the circle. The terminal ray of the angle intersects the circle at point Q. The length of arc PQ is 6 units. Which of the following gives the distance of point Q from the x-axis?
5cos(6/5)
Which of the following expressions is equivalent to log3(x^5)?
5log3(x)
In the xy-plane, two different angles A and B are in standard position and share a terminal ray. Based on this information, which of the following gives possible values for A and B?
A=2pi/3 and B=8pi/3
The figure shows the graph of a function g in the xy-plane with four labeled points. It is known that a relative maximum of g occurs at A, and the only point of inflection of the graph of g is C. Of the following points, at which is the rate of change of g the least?
C
The table gives polar coordinates (r, theta) for four selected points. Which of the points lies in Quadrant II of the xy-plane?
D, because from the positive x-axis, 7pi/4 counterclockwise from the origin is in Quadrant IV, and the negative radius indicates the opposite direction of the angle from the origin.
A set of data is represented using a semi-log plot (not shown), in which the vertical axis is logarithmically scaled. The points on the semi-log plot appear to follow a decreasing linear pattern. Which of the following function types best models the set of data?
Exponential decay
The function C models temperature, in degrees Celsius, as a function of time t, in hours, for t>0. The function P models electricity usage, in kilowatts, as a function of temperature, in degrees Celsius. Let K be the composition function defined by K(t)=P(C(t)). Which of the following statements is true about function K?
K models electricity usage as a function of time.
The table gives values for a rational function f at selected values of x. The polynomial in the numerator and the polynomial in the denominator of the function have no zeros in common. Based on the information given, which of the following conclusions is possible for the graph of f in the xy-plane?
The graph of f has a vertical asymptote at x=-5
The table gives values for a polynomial function g at selected values of x. If a<b, then g(a)>g(b) for all a and b in the interval 3<x<7. Which of the following could be true about the graph of g on the interval 3<x<7?
The graph of g is concave down because the function is decreasing, and the average rate of change over equal-length input value intervals is decreasing.
A regression model S is constructed The residuals from the regression are plotted and labeled. Based on the vertical axis of the residual plot (not shown), point A is located at 1.3 and point B is located at -2.5. Which of the following statements is true about the model and the estimates produced by the model that correspond to A and B?
The model produces an underestimate at A and an overestimate at B. Based on the absolute values of the residuals, there is a greater error in the model with B than with A.
The function r is given by r(x)=x²-x-2/(x+1)²(x-2). In the xy-plane, which of the following is true about holes in the graph of r?
There is a hole at x=2 only because the multiplicity of -1 in the denominator is greater than the multiplicity of -1 in the numerator, and because the multiplicity of 2 in the numerator is equal to the multiplicity of 2 in the denominator.
The figure shows the graph of function g for 0≤x≤13. The endpoints of the interval are labeled with points A and E. Two other extrema for g are labeled with points B and D. Point C is the only point of inflection of the graph of g for 0≤x≤13. Let ta, tb, tc, td, and te represent the x-coordinates at those points. Of the following, on which intervals is the rate of change of g decreasing?
[tc, td] and [td, te]
A portion of the graph of the polar function r=f(theta), where f(theta)=2-4cosx, is shown in the polar coordinate system for a≤x≤b. If 0≤a<b<2pi, which of the following are the values for a and b?
a=0 and b=pi/3
For 0≤t≤16, the rate at which customers arrive at a restaurant on a given day is modeled by the function R, where R(t) is measured in customers per hour and t is measured in hours since the restaurant opened. The function R is increasing for 0<t<4 and 8<t<12, and R is decreasing for 4<t<8 and 12<t<16. The function N models the total number of customers who have arrived at the restaurant since it opened, up to time t. Which of the following could be the graph of y=N(t) for 0≤t≤16?
continually going up, has point (1,0)
In the xy-plane, the graph of a rational function f has a hole at x+2. Input values of f sufficiently close to 2 correspond to output values arbitrarily close to 6. Which of the following could define f(x)?
f(x)=(x-2)(x+4)/(x-2)(x-1)
In the xy-plane, the graph of which the following functions has a vertical asymptote at x=3pi/4?
f(x)=cot(x+pi/4)
The table gives values for a function g at selected values of x. Which of the following statements is true?
g is best modeled by a quadratic function, because the successive 2nd differences of the output values over equal-interval input values are constant.
The function g is given by g(x)=2cos(pix)+1. Which of the following is the graph of g for 0<x<4?
graph with (0,3), (1,-1), and (4,3)
Which of the following statements is true about the exponential function h given by h(x)=-3*4^x?
h is always decreasing, and the graph of h is always concave down.
The amount of water used each day in an office building, measured in hundreds of gallons, is modeled by the function g defined by g(t)=5sin(0.8(t+2))+25, for integer values of t with 0<t<365 days. Using actual data over time, it was determined that the model underestimates the amount of water used each day by 800 gallons. Based on this information, which of the following functions is a better model for the amount of water used each day, measured in hundreds of gallons?
h(t)=5sin(0.8(t+2))+33
In the xy-plane, the function h, given by h(x)=3^(x+2), is a horizontal translation of the exponential function f, given by f(x)=3^x. Which of the following is an equivalent form for h(x) that expresses has a vertical dilation of f?
h(x)=9*3^x
Where both expressions are defined, which of the following is equivalent to sec^2x-1/sec^2x?
sin^2x
For time t hours, 0≤x≤2, the number of people inside a large shopping center is changing at a rate modeled by the function P given by P(t)=t³-4t³+3t+1, where P(t) is measured in hundreds of people per hour. Which of the following gives the time t and reasoning for when the number of people inside the shopping center is at its maximum?
t=1.445, because the rate of change in the number of people inside the shopping center changes from positive to negative.
On a given day, the number of people, in thousands, that have entered a museum is modeled by the function ℎ given by ℎ(t)=4.217tan^-1(0.7t-0.026), where t is measured in hours and 0≤t≤8. Based on the model, at what time t did person number 4000 enter the museum?
t=2.029
The function f is given by f(t)=e^t, and the function g is given by g(t)=7lnt. If the function h is given by h(t)=(f*g)(t), which of the following is an expression for h(t), for t>0?
t^7
At time t=0, water begins pouring into an empty container at a constant rate. The water pours into the container until it is full. The situation is modeled by the given graph, where time, in seconds, is the independent variable and the depth of water in the container, in centimeters, is the dependent variable. For which of the following containers would the graph be appropriate?
wide at bottom, thin in middle, wide at top
The function k is given by k(x)=2sinx. What are all values of x, for 0<x<2pi, where k(x)=-1?
x=7pi/6 and x=11pi/6
