Find the vertical asymptotes and the nonvertical asymptotes for the graph of the function. Do not sketch the graph.
f(x)=x−77x−9 If there are any vertical asymptotes, what are they? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The vertical asymptote(s) is/are x=□ .
(Simplify your answer. Use a comma to separate answers as needed.)
B. There are no vertical asymptotes.
3/17 Caiden deposited $475 in an account that pays an interest rate of 3.8% compounded annually. What will be his balance after 15 years?
\827.52\831.10$839.45
\$846.80
3
Mark for Review Let f be a differentiable function. If h(x)=(1+f(3x))2, which of the following gives a correct process for finding h′(x) ?
(A) h′(x)=2(1+f(3x))
(B) h′(x)=2(1+f(3x))⋅f′(3x)
(C) h′(x)=2(1+f(3x))⋅f′(x) D h′(x)=2(1+f(3x))⋅f′(3x)⋅3
Question?
Find the asymptotes of f(x)=x2+9x+20x+4. Horizontal asymptote at y=1, vertical asy mptotes at x=4,x=5 Horizontal asymptote at y=1, vertical asymptote at x=−4
How many reports do you need to send if 8 board members get 2 copies each and 20 employees get 1 copy?
A. 8+20
B. 8+20+2+1
C. (8+2)×(20+1)
D. (8×2)+(20×1)
E. (20×8)+(2×1)
A person accidentally tosses their cell phone into the air while standing near the base of the CN Tower. The height h(t) of the phone (in meters above the ground) after t seconds is modelled by:
h(t)=−at2+4at+2, where a∈R What is the expression for the instantaneous rate of change of the phone's height a t=1 second?
2a
−2a+6
5
3a+2
20. People are entering a stadium at a steady rate of 32 people per minute. When the gates open, there are already 46 people in the stadium. No one leaves the stadium for the first hour after the gates have opened.
(a) How many people will be in the stadium 30 minutes after it opens? Show the calculations that lead to you answer.
(b) Write a linear equation for the number of people, n, as a function of the time in minutes, m, since the gates were opened.
32(30)=960 people +46 people 1006 people n(m)=32n+46
(c) After one hour, no additional people enter, but some start to leave. If it takes a total of 4 hours for the stadium to completely empty, what is the average rate at which people leave, in people per hour? Show the calculations that lead to your answer.
h(60)=32(60)+46=1966 prople =1966 people
31966=6553⩾ people pe
78. If f(x)=ln(x) and g is a differentiable function with domain x>0 such that limx→∞g(x)=∞ and g′ has a horizontal asymptote at y=4 then limx→∞g(x)f(x) is
A. 0
B. -4
C. 4
D. nonexistent
Find the horizontal asymptote, if any, of the graph of the rational function.
h(x)=5x2+811x3 Select the correct choice below and, if necessary, fill in the answer box to complete your choi
A. The horizontal asymptote is □ . (Type an equation.)
B. There is no horizontal asymptote.
Find the horizontal asymptote, if any, of the graph of the rational function.
f(x)=5x+4−6x+7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The horizontal asymptote is □ .
(Type an equation. Simplify your answer. Use integers or fractions for any numbers in the equation.)
B. There is no horizontal asymptote.
Essay 10 points
Let f(x)=x+51−4
- Enter the equation of the vertical asymptote of f(x).
- Enter the equation of the horizontal asymptote of f(x).
- Enter the domain of f(x) in interval notation.
- Enter the range of f(x) in interval notation.
- State the vertical and/or horizontal transformation(s).
- Graph f(x), by hand.
- Include:
- Asymptotes as dotted lines
- x-intercepts as precise points
- Open dot where there is a hole in the graph (if applicable)
(1 point) each correct answer
(5 points) correct labeled graph
Remember, you can type in your answer and work or you can take a picture of it and upload the image.
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The mass of a colony of bacteria, in grams, is modeled by the function P given by P(t)=2+5tan−1(2t), where t is measured in days. What is the instantaneous rate of change of the mass of the colony, in grams per day, at the moment the colony reaches a mass of 6 grams?
(A) -0.606
(B) 0.250
(C) 1.214
(D) 1.942
Find the slope of the line that passes through each set of points: (4,4) and (6,6). Find the answer below and decorate the stocking with the features listed.
90. What is the absolute minimum value of y=34x3−8x2+15x on 1≤x≤3 ?
A. 0
f(x)1=4x2−16x+15
B. 325
C. 9
f(1)=3251(3)=94x2−10x−6x+152x(2x−5)−3(2x−5)
D. 352f(3)=9f∣5)=3252x(2x−5)−3(2x−5(2x−3)(2x−5)
A bug has fallen into a whirlpool, and it's distance from the center is given by: r=θ, where 0≤θ≤4π It is being sucked from the outside towards the center of the swirl in the whirlpool. What is the horizontal component of the bug's location, after it has spun through an angle of 611π radians?
x=−4.68x=5.89x=4.99x=9.56
For time 0≤t≤10, water is flowing into a small tub at a rate given by the function F defined by F(t)=arctan(2π−10t). For time 5≤t≤10, water is leaking from the tub at a rate given by the function L defined by L(t)=0.03(20t−t2−75). Both F(t) and L(t) are measured in cubic feet per minute, and t is measured in minutes. The volume of water in the tub, in cubic feet, at time t minutes is given by W(t).
(a) At time t=3, there are 2.5 cubic feet of water in the tub. Write an equation for the locally linear approximation of W at t=3, and use it to approximate the volume of water in the tub at time t=3.5. No response entered
(b) Find W′′(8). Using correct units, interpret the meaning of W′′(8) in the context of the problem. No response entered
(c) Is there a time t, for 5<t<10, at which the rate of change of the volume of water in the tub changes from positive to negative? Give a reason for your answer.
Find an equation for a sinusoidal function that has period 2π, amplitude 2 , and contains the point ( 2π,0 ). Write your answer in the form f(x)=Asin(Bx+C)+D, where A,B,C, and D are real numbers.
f(x)=□
Let f be a twice-differentiable function such that f′(1)=0. The second derivative of f is given by f′′(x)=x2cos(x2+π) for −1≤x≤3.
(a) On what open intervals contained in −1<x<3 is the graph of f concave up? Give a reason for your answer. No response entered
(b) Does f have a relative minimum, a relative maximum, or neither at x=1 ? Justify your answer. No response entered
(c) Use the Mean Value Theorem on the closed interval [−1,1] to show that f′(−1) cannot equal 2.5. No response entered
(d) Does the graph of f have a point of inflection at x=0 ? Give a reason for your answer.
A particle moves along the x-axis so that its position at time t>0 is given by x(t)=3t2+8t2−9.
(a) Show that the velocity of the particle at time t is given by v(t)=(3t2+8)270t. No response entered
(b) Is the particle moving toward the origin or away from the origin at time t=2 ? Give a reason for your answer. No response entered
(c) The acceleration of the particle is given by a(t). Write an expression for a(t), and find the value of a(2). No response entered
(d) What position does the particle approach as t approaches infinity?
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
W(t)={217−23cos(6πt)10−51(t−6)2 for 0≤t≤6 for 6<t≤10 The depth of a river at a certain point is modeled by the function W defined above, where W(t) is measured in feet and time t is measured in hours.
(a) Find W′(8). Using correct units, explain the meaning of W′(8) in the context of the problem. No response entered
(b) The graph of W is concave down for 3≤t≤3.5. Use the line tangent to the graph of W at t=3 to show that W(3.5)≤9. No response entered
(c) Find limt→2t−2w(t)−t3+41.
\begin{tabular}{|c||c|c|c|c|}
\hlinex & -3 & -2 & -1 & 1 \\
\hlinef(x) & −25 & -3 & -2 & 32 \\
\hlinef′(x) & -1 & 31 & 56 & 34 \\
\hline
\end{tabular} The table above gives values of the differentiable function f and its derivative for selected values of x.
(a) Let g be the function defined by g(x)=exf(x2). Find g′(−1). No response entered
(b) Let h be the function defined by h(x)=f(f(−2x)). Find h′(1). No response entered
(c) Let k be the function defined by k(x)=f(x)⋅arcsin(2x). Find k′(−1).
Use logarithmic differentiation to differentiate each function with respect to x. You do not need to simplify or substitute for y.
y=(2x4−1)5⋅(4x9+5)6(x2+4)3
A) dxdy=y(x2+46x−2x4−1160x3−4x9+5864x8)
B) dxdy=y(x2+418x−2x4−180x3+4x9+5432x8)
C) dxdy=y(x2+46x−2x4−140x3+4x9+5864x8)
D) dxdy=y(x2+46x−2x4−140x3−4x9+5216x8)
Let g be the function defined by g(x)=(x2−x+1)ex. What is the absolute maximum value of g on the interval [−4,1]?
(A) 1
(B) e
(C) e3
(D) e121
https://apclassroom.collegeboard.org/25/assessments/results/62905152/performance/591...
The graph of f′′, the second derivative of the continuous function f, is shown above on the interval [0,9]. On this interval f has only one critical point, which occurs at x=6. Which of the following statements is true about the function f on the interval [0,9] ?
(A) f has a relative minimum at x=6 but not an absolute minimum.
(B) The absolute minimum of f is at x=6. C f has a relative maximum at x=6 but not an absolute maximum.
(D) The absolute maximum of f is at x=6.
Find the derivative.
y=e6x+7ln(6x+7) Select one:
A. ln[6x+7]e(6x+7)1−6[ln(6x+7)]2
B. (6x+7)e(6x+7)1
C. (6x+7)e(6x+7)6−(36x+42)ln(6x+7)
D. (6x+7)e(6x+7)1−(6x+7)ln(6x+7)
23-28 True-False Determine whether tr - statement is true or false. Explain your answer. 23. If f(x) is continuous at x=c, then so is ∣f(x)∣. 24. If ∣f(x)∣ is continuous at x=c, then so is f(x). 25. If f and g are discontinuous at x=c, then so is f+g. 26. If f and g are discontinuous at x=c, then so is fg.
Rational Functions, Equations, and Inequalities NAME Williams 0. DATE :
181
1212024
K 1/16
T
15
c
14
A
112
1) For each function, identify the location of the hole (if applicable), the equation(s) of the vertical asymptote(s) and the equation of the horizontal asymptote.
(a) f(x)=x2+xx+1
(b) g(x)=x2−4x2+4x+4
[K-6]
The acceleration, in meters per second per second, of a race car is modeled by A(t)=t3−215t2+12t+10, where t is measured in seconds. What is the car's maximum acceleration on the time interval 0≤t≤6 ?
(A) The maximum acceleration of the race car is 2 meters per second per second and occurs at t=4 seconds. B The maximum acceleration of the race car is 6 meters per second per second and occurs at t=28 seconds.
C. The maximum acceleration of the race car is 15.5 meters per second per second and occurs at t=1 second.
(D) The maximum acceleration of the race car is 28 meters per second per second and occurs at t=6 seconds.
6) Explain how a quadratic function and its reciprocal function are related with regards to positive and negative intervals. Use an example with your explanation.
7) A biologist predicted that the population of tadpoles in a pond could be modelled by the function f(x)=x+740x, where x is given in days and 0≤x≤10. The function that actually models the tadpole population is g(x)=(x+7)(x+1)80 for 0≤x≤10. Determine when f(x)≥g(x).
Mitosis is a process of cell reproduction in which one cell divides into two identical cells. E, coli is a fast-growing bacterium that is often responsible for food poisoning in uncooked meat. It can reproduce itself in 15 minutes. If you begir with 100 E. coli bacteria, how many will there be in 1 hour?
a. 1200 bacteria
c. 1500 bacteriá
b. 1400 bacteria
d. 1600 bacteria Please select the best answer from the choices provided
A
B
C
D
Derivatives of Inverse Trig Functions
Score: 0/1
Penalty: none Question
Watch Video If f(x)=sin−1(x), then what is the value of f′(54) in simplest form? Answer Attempt 1 out of 5
□
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Graph of y=f(x)
Consider the function f(x)=−(2x+16)(x−7)2 with restricted domain (−9,9) which is graphed above and let g(x) be defined as g(x)=∣f(x)∣.
a) Find the x-coordinates of the local extrema of f(x) in the open interval ( −9,9). Enter "none" (without the quotation marks) if there is none. Local maximum at x=7
Local minimum at x=−3
b) Find the x-coordinate(s) of any local extrema of g(x) in the open interval (−9,9). If more than one, separate with semicolon(s) and if none then enter "none" (without the quotation marks).
Local maximum at x=−3
Local minimum at x=−8;7
c) Find the open interval(s) on which the graph of y=g(x) is concave down. Enter your answer in interval notation such as ( a,b ). If more than one interval, instead of using the union symbol, separate the intervals with a comma (i.e (a,b), (c,d)).
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Graph of y=f(x)
Consider the function f(x)=−(2x+10)(x−4)2 with restricted domain (−6,6) which is graphed above and let g(x) be defined as g(x)=∣f(x)∣.
a) Find the x-coordinates of the local extrema of f(x) in the open interval (−6,6). Enter "none" (without the quotation marks) if there is none. Local maximum at x= Number
□
Local minimum at x= Number
□ Section Attempt 1 of 2
Grade 7 Math Unit 3 Assessment 24-25
Question 2
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Question
Normal The number of gallons of water in a tank, y, over a period of x hours is shown in the graph below. What is the constant of proportionality in this situation?
A. 5 gallons per hour
For each table, determine whether it shows a direct variation. If it does, write its direct variation equation.
\begin{tabular}{|c|c|}
\hlinex & y \\
\hline 4 & 1 \\
\hline 12 & 3 \\
\hline 20 & 5 \\
\hline
\end{tabular}
Not direct variation
\begin{tabular}{|c|c|}
\hlinex & y \\
\hline 2 & 1 \\
\hline 5 & 2.5 \\
\hline 9 & 4.5 \\
\hline
\end{tabular}
Not direct variation =
Direct variation
Direct variation Equation: Equation:
□
4. SpongeBob wants to go to point D from point A on an island. He can swim to any point C on the beach. He can swim at 4km/hr and run at 5km/hr. (a) Find analytically the location of C between B and D that will take the least amount of time.
(b) Find the time it would take to swim from A to C and then run from C to D using the result of
)
(c) Find the time it would take if Spongebob swam from A to B, and then run from B to D
(d) Find the time if Spongebob swam directly from A to D, and compare the results with those of (b) and (c).
3. A tea kettle is taken off of the stove and is cooling on the countertop for 10 minutes. H′(t), a differentiable function, represents the rate at which the temperature is changing, measured in degrees Celsius per minute, and t is measured in minutes.
\begin{tabular}{|c|c|c|c|c|c|}
\hlinet(min) & 0 & 2 & 5 & 9 & 10 \\
\hlineH′(t)(∘C/min) & -2.1 & -1.8 & -1.6 & -1.2 & -0.8 \\
\hline
\end{tabular}
(c) If the temperature of the tea in the kettle was 96∘C when it was taken off the stove, what is the temperature after 10 minutes?
The value of an investment (in dollars) after t years is gives by
A(t)=100(1.03)t Find the average rate of change of the value (in dollars per year) over the first 5 years, that is, on the interval [0,5]. Round to the nearest cent, and do not include the units or a dollar sign; just type in a qumber.
6.) Josie determines that she can only afford a car payment of $250 per month. The car she wants to purchase has a 4.22\% APR for 60 months and a down payment of $500. The dealership calculates a monthly payment of $350 What are some things that will lower Josie's monthly payment?
Increase her dain pagment and find a laver mitrest rate frama tre car longer
Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function
D′(t)=848.18+816.08t−151.95t2+17.76t3
where t is the number of years since 1995. By how much did the debt increase between 1996 and 2007? The debt increased by $72,270.55 billion.
(Round to two decimal places as needed.)
Question 8 (1 point)
Which of the following functions has exactly one vertical asymptote when graphed?
A) f(x)=x2−3x6
B) f(x)=x2−3x6x
C) f(x)=x6x2−18x
D) B and C