6/4 Exam

In the 𝑥𝑦-plane, the function ℎ is graphed on the interval 0≤𝑥≤10. The point 3,-2 is on the graph of 𝑦=ℎ𝑥. Which of the following points is on the graph of 𝑦=2ℎ𝑥-1 ? (3,-2) (3,-5) (5,-2) (6,-3)

(3,-5)

The function 𝐶 models the cost, in dollars, for producing 𝑥 items and is given by 𝐶𝑥=1000+𝑏𝑥𝑥, 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 -$5 per item -$8.33 per item -$50 per item

-$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 -1/2 2 13

-2

The figure shows a circle centered at the origin with an angle of measure 𝜃 radians in standard position. The terminal ray of the angle intersects the circle at point 𝑃. If tan𝜃=-34, what is the slope of the line that passes through the origin and 𝑃 ? -4/3 -4/5 -3/4 -3/5

-3/4

An object travels along a horizontal line with velocity given by the function 𝑣. The graph of 𝑦=𝑣𝑡, which consists of four line segments, is shown for 0≤𝑡≤10. On which of the following intervals is the object's velocity decreasing over the entire interval? 0 < 𝑡 < 4 5 < 𝑡 < 8 8 < 𝑡 < 10 only 0 < 𝑡 < 3 and 8 < t < 10

0 < 𝑡 < 3 and 8 < t < 10

The polar function 𝑟=𝑓𝜃, where 𝑓𝜃=1+2cos𝜃, is graphed in the polar coordinate system for 0≤𝜃≤2𝜋. On which of the following intervals of 𝜃 is the distance between the point with polar coordinates 𝑓𝜃,𝜃 and the origin decreasing? (0,2.094) only (2.094,4.189) (0,2.094) and (3.142,4.189) (2.094,3.142) and (4.189, 6.283)

0,2.094) and (3.142,4.189)

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 ? 54.2 127.0 309.3 942.1

127.0

The function s is given by S(t) = 500,000/1+0.4e^(kt), where k is constant. If S(4)=300,000, what is the value of S(12)? 175,325 214,772 343,764 357,143

175,325

The figure shows a square sheet of cardboard on the left and an open box on the right. The square sheet has side length 12 inches (in) and is used to make the open box by removing from each of the four corners a square of side length 𝑥 inches, and folding up the sides. For what value of 𝑥 does the box have the maximum possible volume? Note: The volume of a rectangular solid with length 𝑙, width 𝑤, and height ℎ is 𝑉=𝑙𝑤ℎ. 2 3 4 6

2

The figure shows the graph of a quartic polynomial function with zeros at 𝐴, 𝐵, and 𝐶. Which of the following statements gives the multiplicity of the zero at 𝐶 with correct reasoning? 1, because the graph of the function intersects the 𝑥-axis once at 𝐶. 1, because the quartic polynomial function has four zeros: exactly one at 𝐴, one at 𝐵, one at 𝐶, and a non-real zero. 2, because the quartic polynomial function has three real zeros, one of which must have a multiplicity of 2, and the polynomial does not change signs at 𝐶. 2, because 𝐶 is the second zero to the right of the origin, and there are more zeros to the right than to the left of the origin.

2, because the quartic polynomial function has three real zeros, one of which must have a multiplicity of 2, and the polynomial does not change signs at 𝐶.

Consider the sinusoidal functions 𝑓 and 𝑔, defined by 𝑓𝑥=3cos𝑥+2.78 and 𝑔𝑥=sin𝑥-0.25. Which of the following describes a zero of the function ℎ defined by ℎ𝑥=𝑓𝑥-𝑔𝑥 on the interval 0,𝜋 ? 2.242 1.932 1.601 0.250

2.242

The polynomial function 𝑓 is an odd function with domain -4≤𝑥≤4. The table gives information about values of 𝑓𝑥 and the behavior of the function. What is the absolute maximum value of 𝑓 on its domain? 4 3 2 0

3

A builder is trying to find the maximum area that can be bounded by a rectangular region on a certain plot of land. The builder determined four different areas based on the length 𝑥 of one side of the rectangle, for four different rectangles. The table gives values of the one side of the rectangle, in meters (m), and the corresponding area, in square meters (m2). After noticing that as 𝑥 increases, the area values increase and then decrease, the builder used a quadratic regression to produce a function 𝐴 that models the area based on the constraints. Of the following, what is the maximum value of 𝐴𝑥, to the nearest integer, based on the model? 343 347 354 362

347

The graph of the piecewise-linear function 𝑓 is shown in the figure. Let 𝑔 be the inverse function of 𝑓. What is the maximum value of 𝑔 ? ​1/7 ​1/5 ​5 7

5

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? 5073 5203 5333 5345

5345

The average monthly high temperature, in degrees Fahrenheit (°F), varies periodically for a certain city. The table gives the average monthly high temperature for selected months of the year. Based on a sinusoidal regression of these data, a model is constructed to predict average monthly high temperature as a function of time 𝑡, the number of the month of the year. What is the average monthly high temperature predicted by the model for June 𝑡=6 ? 53.7°F 61.8°F 64.9°F 65.3

64.9°F

The function 𝐿 is defined by 𝐿𝑡=𝑎𝑒-0.0149𝑡+𝑏, where 𝑎 and 𝑏 are positive constants. If 𝐿0=15 and lim𝑡→∞𝐿𝑡=1, for what value of 𝑡 is 𝐿𝑡=6 ? 56.866 61.496 69.102 73.732

69.102

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? Linear Exponential growth Exponential decay Logarithmic

Exponential decay

In the polar coordinate system, the graph of a polar function 𝑟=𝑓𝜃 is shown with a domain of all real values of 𝜃 for 0≤𝜃<2𝜋. On this interval of 𝜃, the graph has no holes, passes through each point exactly one time, and as 𝜃 increases, the graph passes through the labeled points 𝐴, 𝐵, 𝐶, and 𝐷, in that order. On which of the following intervals is the average rate of change of 𝑟 with respect to 𝜃 greatest? From 𝐴 to 𝐶 From 𝐵 to 𝐶 From 𝐵 to 𝐷 From 𝐶 to 𝐷

From 𝐶 to 𝐷

The figure shows a swimming pool filled with water. A pump is used to remove water from the pool until the pool is empty. When the pump is running, the rate at which the volume of water in the pool decreases is constant. Graph: (10, 0) Graph: (6, 0) High y-axis points slightly apart Graph: (6, 0) Mid y-axis Graph: (6, 0) High y-axis points closer together

Graph: (6, 0) High y-axis points slightly apart

For 0≤𝑡≤16, the rate at which customers arrive at a restaurant on a given day is modeled by the function 𝑅, where 𝑅𝑡 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=𝑁𝑡 for 0≤𝑡≤16? Shaped like M Shaped like W Linear slant upward ends concave up Linear slant upward ends concave down

Linear slant upward ends concave down

Consider the piecewise function 𝑊 defined by 𝑊𝑡=125-0.2𝑡2for 0≤𝑡<20445-20𝑡for 20≤𝑡≤22. Which of the following is a correct statement about the average rates of change of 𝑊 ? The average rate of change of 𝑊 for 0≤𝑡≤5 is less than the average rate of change of 𝑊 for 5≤𝑡≤10. The average rate of change of 𝑊 for 10≤𝑡≤15 is less than the average rate of change of 𝑊 for 5≤𝑡≤10. The average rate of change of 𝑊 for 15≤𝑡≤20 is greater than the average rate of change of 𝑊 for 10≤𝑡≤15. The average rate of change of 𝑊 for 21≤𝑡≤22 is less than the average rate of change of 𝑊 for 20≤𝑡≤21.

The average rate of change of 𝑊 for 10≤𝑡≤15 is less than the average rate of change of 𝑊 for 5≤𝑡≤10.

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 an 𝑥-intercept at 𝑥=-5. The graph of 𝑓 has a hole at 𝑥=-5. The graph of 𝑓 has a vertical asymptote at 𝑥=-5. The graph of 𝑓 has a horizontal asymptote at 𝑦=-5.

The graph of 𝑓 has a vertical asymptote at 𝑥=-5.

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? [𝑡𝐴,𝑡𝐵] only [𝑡𝐷,𝑡𝐸] only [𝑡𝐴,𝑡𝐵] and [𝑡𝐷,𝑡𝐸] [𝑡𝐶,𝑡𝐷] and [𝑡𝐷,𝑡𝐸]

[𝑡𝐶,𝑡𝐷] and [𝑡𝐷,𝑡𝐸]

​The function f is defined by f(x) = asin(b(x+c)) + d, for constants a, b, c, and d. In the xy plane the points (2,2) and (4,4) represent a minimum value and a maximum value, respectively, on the graph of f. What are the values of a and d? a=1 d=3 a=1 d=2 a=2 d=3 a=2 d=2

a=1 d=3

The rate of people entering a subway car on a particular day is modeled by the function R a. t=20 b. t=17.056 c. t=13.295 d. t=5.505

t=5.505

The table gives values for a function m at selected values of t. Which of the following graphs could represent these data in a semi-log plot, where the vertical axis is logarithmically scaled? y starts around 25 y starts around 1.5 y starts around 50 y starts 1.5

y starts around 50

​The function 𝑓 is given by 𝑓𝑥=sin2.25𝑥+0.2. The function 𝑔 is given by 𝑔𝑥=𝑓𝑥+0.5. What are the zeros of 𝑔 on the interval 0≤𝑥≤𝜋 ? ​1.085 and 2.481 ​1.307 and 2.704 ​1.540 and 2.471 ​0.144, 1.075, and 2.936

​1.540 and 2.471

​The function 𝑓 is given by 𝑓𝑥=2sin4𝑥+cos2𝑥. Using the period of 𝑓, which of the following is the number of complete cycles of the graph of 𝑓 in the 𝑥𝑦-plane on the interval 0≤𝑥≤1000 ? ​159 ​318 ​602 636

​318

At a coastal city, the height of the tide, in feet (ft), is modeled by the function h ​The maximum height of the tide is 13.8 ft The maximum height of the tide occurs at t=6 hours ​The minimum height of the tide is 1 ft The minimum height of the tide occurs at t=12 hours

​The maximum height of the tide is 13.8 ft

​Two function models 𝑘 and 𝑚 are constructed to represent the sales of a product at a group of grocery stores. Both 𝑘𝑡 and 𝑚𝑡 represent the sales of the product, in thousands of units, after 𝑡 weeks for 𝑡≥2. If 𝑘𝑡=14-2.885ln𝑡 and 𝑚𝑡=-𝑡+14, what is the first time 𝑡 that sales predicted by the logarithmic model will be 0.1 thousand units more than sales predicted by the linear model? 𝑡=6.318 ​𝑡=4.324 ​𝑡=3.577 ​𝑡=2.289​

​𝑡=4.324

The figure shows a portion of the graph of a function 𝑓. Which of the following conclusions is possible for 𝑓 ? 𝑓 is a quadratic function because the output values are proportional over equal-length input-value intervals. 𝑓 is a quadratic function because the average rates of change over consecutive equal-length input-value intervals can be given by a linear function. 𝑓 is an exponential function because the output values are proportional over equal-length input-value intervals. 𝑓 is an exponential function because the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.

𝑓 is an exponential function because the output values are proportional over equal-length input-value intervals.

A large wheel of radius 2 feet is rotated at a constant rate. The figure provides a representation of the wheel in the 𝑥𝑦-plane with the direction of rotation indicated. At time 𝑡=0 minutes, the wheel begins to rotate. Point 𝑃 on the wheel is at the "Start" position in the figure. At time 𝑡=20 minutes, 120 rotations of the wheel have been completed, and 𝑃 is in the same position as it was at time 𝑡=0. A sinusoidal function is used to model the 𝑦-coordinate of the position of 𝑃 as a function of time 𝑡 in minutes. Which of the following functions is an appropriate model for this situation? ​𝑓𝑡=2sin𝜋(10/𝑡) ​𝑓𝑡=2sin(𝜋/3𝑡) ​𝑓𝑡=2sin(6𝑡​) 𝑓𝑡=2sin(12𝜋t)

𝑓𝑡=2sin(12𝜋t)

For time 𝑡 hours, 0≤𝑡≤2, the number of people inside a large shopping center is changing at a rate modeled by the function 𝑃 given by 𝑃𝑡=𝑡3-4𝑡2+3𝑡+1, where 𝑃𝑡 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=0.451, because the rate of change in the number of people inside the shopping center changes from increasing to decreasing. 𝑡=0.451, because the rate of change in the number of people inside the shopping center changes from positive to negative. 𝑡=1.445, because the rate of change in the number of people inside the shopping center changes from increasing to decreasing. 𝑡=1.445, because the rate of change in the number of people inside the shopping center changes from positive to negative.

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 ℎ𝑡=4.217tan-10.7𝑡-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=1.121 𝑡=2.029 𝑡=5.165 6.623

𝑡=2.029