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mathematics
calculus

Questions and Answers of Calculus

Determine which of the following functions f(x) = cx, g(x) = cx2, h(x) = c (|x|, and r(x) c/x can be used to model the data and determine the value of the constant c that will make the function fit
Determine whether the statement is true or false. Justify your answer. 1. Every relation is a function. 2. Every function is a relation. 3. For the function f(x) = x4 1 The domain is (- (, () and the
Describe the error.The functionsf(x) = (x 1 and g(x) = 1 / (x - 1Have the same domain, which is the set of all real numbers x such that x ( 1?
Which sets of ordered pairs represent functions from A to B? Explain. 1. A = {0, 1, 2, 3} and B = {−2, −1, 0, 1, 2} (a) {(0, 1), (1, −2), (2, 0), (3, 2)} (b) {(0, −1), (2, 2), (1, −2), (3,
Consider f(x) = (x - 2 and g(x) = 3(x - 2. Why are the domains of f and g different?
1. Given f(x) = x2, is f the independent variable? Why or why not?2. The graph represents the height h of a projectile after t seconds.Time (in seconds) (a) Explain why h is a function of t. (b)
(a) The sales tax on a purchased item is a function of the selling price. (b) Your score on the next algebra exam is a function of the number of hours you study the night before the exam. Determine
(a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped. Determine
Use the Vertical Line Test to determine whether the graph represents y as a function of x. To print an enlarged copy of the graph, go to MathGraphs.com.1.2. 3. 4.
Find the zeros of the function algebraically. 1. f(x) = 3x + 18 2. f(x) = 15 − 2x 3. f(x) = 2x2 − 7x − 30 4. f(x) = 3x2 + 22x − 16
(a) Use a graphing utility to graph the function and find the zeros of the function and (b) Verify your results from part (a) algebraically. 1. f(x) = x2 − 6x 2. f(x) = 2x2 − 13x − 7 3. f(x) =
Determine the open intervals on which the function is increasing, decreasing, or constant.1. f(x) = ½ x32. f(x) = x2 - 4x 3. f(x) = (x2 - 1 4. f(x) = x3 - 3x2 + 2
Use a graphing utility to graph the function and visually determine the open intervals on which the function is increasing, decreasing, or constant. Use a table of values to verify your results. 1.
Use a graphing utility to approximate (to two decimal places) any relative minima or maxima of the function? 1. f(x) = x(x + 3) 2. f(x) = −x2 + 3x − 2 3. h(x) = x3 − 6x2 + 15 4. f(x) = x3 −
Graph the function and determine the interval(s) for which f(x) ( 0? 1. f(x) = 4 − x 2. f(x) = 4x + 2 3. f(x) = 9 − x2 4. f (x) = x2 − 4x
Find the average rate of change of the function from x1 to x2. Function x-Values Function _________________________ x-value 1. f(x) = −2x + 15 ...................... x1 = 0, x2 = 3 2. f(x) = x2 −
The amounts (in billions of dollars) the U.S. federal government spent on research and development for defense from 2010 through 2014 can be approximated by the model y = 0.5079t2 − 8.168t + 95.08
Use the information in Example 7 to find the average speed of the car from t1 = 0 to t2 = 9 seconds. Explain why the result is less than the value obtained in part(b) of Example7. Refer to Example
An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. t1 = 0, t2 = 3? (a) Use the position equation s 16t2 + v0t + s0 to write a function that represents the
An object is thrown upward from a height of 6.5 feet at a velocity of 72 feet per second. t1 = 0, t2 = 4? (a) Use the position equation s 16t2 + v0t + s0 to write a function that represents the
An object is thrown upward from ground level at a velocity of 120 feet per second. t1 = 3, t2 = 5? (a) Use the position equation s 16t2 + v0t + s0 to write a function that represents the situation,
Use the graph of the function to find the domain and range of f and each function value.1. (a) f (ˆ’1)(b) f(0)(c) f(1)(d) f(2)2. (a) f (ˆ’1) (b) f (0) (c) f (1) (d) f(3) 3. (a) f
An object is dropped from a height of 80 feet. t1 = 1, t2 = 2? (a) Use the position equation s 16t2 + v0t + s0 to write a function that represents the situation, (b) Use a graphing utility to graph
Determine whether the function is even, odd, or neither. Then describe the symmetry. 1. f(x) = x6 2x2 + 3 2. g(x) = x3 5x 3. h(x) = x(x + 5 4. f(x) = x(1 − x2
Sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. 1. f(x) = −9 2. f(x) = 5 − 3x 3. f(x) = −|x - 5| 4. h(x) = x2 - 4
Write the height h of the rectangle as a function of x.1.2.
Write the length L of the rectangle as a function of y.1.2.
Describe the error.The function f(x) = 2x3 - 5 is odd because f(ˆ’x) = ˆ’f(x), as follows.f(ˆ’x) = 2(ˆ’x)3 ˆ’ 5= ˆ’2x3 ˆ’ 5=
Corners of equal size are cut from a square with sides of length 8 meters (see figure).(a) Write the area A of the resulting figure as a function of x. Determine the domain of the function. (b) Use a
Each function described below models the specified data for the years 2006 through 2016, with t = 6 corresponding to 2006. Estimate a reasonable scale for the vertical axis (e.g., hundreds,
The table shows the temperatures y (in degrees Fahrenheit) in a city over a 24-hour period. Let x represent the time of day, where x = 0 corresponds to 6A.M. Time, x ______________ Temperature, y 0
1. A function with a square root cannot have a domain that is the set of real numbers? 2. It is possible for an odd function to have the interval [0, () as its domain. 3. It is impossible for an even
Use the graph of the function to answer parts (a)-(e).(a) Find the domain and range of f. (b) Find the zero(s) of f. (c) Determine the open intervals on which f is increasing, decreasing, or
Find the coordinates of a second point on the graph of a function f when the given point is on the graph and the function is (a) even and (b) odd. 1. (-5/3, -7) 2. (2a, 2c)
Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) y = x (b) y = x2 (c) y = x3 (d) y = x4 (e) y = x5 (f) y
Graph each of the functions with a graphing utility. Determine whether each function is even, odd, or neither. f(x) = x2 − x4 ........................... g(x) = 2x3 + 1 h(x) = x5 − 2x3 + x
Determine whether g is even, odd, or neither when f is an even function. Explain. (a) g(x) = −f(x) (b) g(x) = f(−x) (c) g(x) = f (x) - 2 (d) g(x) = f (x − 2)
Write the most specific name of the function. 1. f (x) = [[x]] 2. f(x) = x 3. f (x) = 1/x 4. f(x) = x2 5. f(x) = (x 6. f(x) = c 7. f(x) = | x | 8. f(x) = x3 9. f(x) = ax + b 10. Fill in the blank:
(a) Write the linear function f that has the given function values and (b) Sketch the graph of the function. 1. f(1) = 4, f(0) = 6 2. f(−3) = −8, f(1) = 2 3. f(1/2) = −5/3, f(6) = 2 4. f(3/5) =
Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. 1. f(x) = 2.5x − 4.25 2. f(x) = 5/6 - 2/3x 3. g(x) = x2 + 3 4. f(x) = −2x2 − 1
Evaluate the function for the given values. 1. f(x) = [[x]] (a) f(2.1) (b) f(2.9) (c) f(−3.1) (d) f(1/2) 2. h(x) = [[x + 3]] (a) h(−2) (b) h(1/2) (c) h(4.2) (d) h(−21.6) 3. k(x) = [[2x +
Sketch the graph of the function? 1. g(x) = − [[x]] 2. g(x) = 4[[x]] 3. g(x) = [[x]] - 1 4. g(x) = [[x - 3]]
Sketch the graph of the function.1.2. 3. 4.
(a) Use a graphing utility to graph the function and (b) State the domain and range of the function? 1. s(x) = 2 (1/4x [[1/4x]]) 2. k(x) = 4 (1/2x [[1/2x]])2
A mechanic's pay is $14 per hour for regular time and time-and-a-half for overtime. The weekly wage function isWhere h is the number of hours worked in a week. (a) Evaluate W(30), W(40), W(45), and
The table shows the monthly revenue y (in thousands of dollars) of a landscaping business for each month of the year 2016, with x = 1 representing January.Month, x ______________ Revenue, y1
The intake pipe of a 100-gallon tank has a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows the volume V of fluid in the tank as a
The cost of mailing a package weighing up to, but not including, 1 pound is $2.72. Each additional pound or portion of a pound costs $0.50. (a) Use the greatest integer function to create a model for
During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour.
For each graph of f shown below, answer parts (a)-(d).(a) Find the domain and range of f. (b) Find the x- and y-intercepts of the graph of f. (c) Determine the open intervals on which f is
A piecewise-defined function will always have at least one x-intercept or at least one y-intercept. Determine whether the statement is true or false. Justify your answer?
A linear equation will always have an x-intercept and a y-intercept. Determine whether the statement is true or false. Justify your answer?
1. Use the graph of f(x) = x2 to write an equation for the function represented by each graph.a.b. 2. Use the graph of f (x) = x3 to write an equation for the function represented by each graph.
Use the graph of f (x) = x3 to write an equation for the function represented by each graph.a.b.
Use the graph of f (x) = |x| to write an equation for the function represented by each graph.a.b.
Use the graph of f (x) = ˆšx to write an equation for the function represented by each graph.a.b.
1.2. 3. 4. Identify the parent function and the transformation represented by the graph. Write an equation for the function represented by the graph.
g is related to one of the parent functions described in Section 2.4. (a)Identify the parent function f. (b)Describe the sequence of transformations from f to g. (c) Sketch the graph of g. (d)
1. The shape of f (x) = x2, but shifted three units to the right and seven units down 2. The shape of f (x) = x2, but shifted two units to the left, nine units up, and then reflected in the x-axis 3.
Match each function h with the transformation it represents, where c > 0. (a) h(x) = f (x) + c (i) A horizontal shift of f, c units to the right (b) h(x) = f (x) - c (ii) A vertical shift of f, c
Use the graph of f (x) = x2 to write an equation for the function represented by each graph.a.b.
Use the graph of f(x) = x3 to write an equation for the function represented by each graph.a.b.
Use the graph of f (x) = |x| to write an equation for the function represented by each graph.a.b.
For each function, sketch the graphs of the function when c = −2, −1, 1, and 2 on the same set of coordinate axes. (a) f(x) = |x| + c (b) f(x) = |x − c|
Use the graph of f (x) = ˆšx to write an equation for the function represented by each graph.a.b.
For each function, sketch the graphs of the function when c = −3, −2, 2, and 3 on the same set of coordinate axes. (a) f(x) = √x + c (b) f(x) = √x - c
The horsepower H required to overcome wind drag on a particular automobile is given byH(x) = 0.00004636x3where x is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the
N (in millions) of households in the United States from 2000 through 2014 can be approximated by N(x) = −0.023(x − 33.12)2 + 131, 0 ≤ t ≤ 14 where t represents the year, with t = 0
1. The graph of y = f (− x) is a reflection of the graph of y = f (x) in the x-axis. 2. The graph of y = − f (x) is a reflection of the graph of y = f (x) in the y-axis. 3. The graphs of f (x) =
The graph of y = f (x) passes through the points (0, 1), (1, 2), and (2, 3). Find the corresponding points on the graph of y = f (x + 2) − 1.
Two methods of graphing a function are plotting points and translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function? Explain. (a) f(x) =
For each function, sketch the graphs of the function when c = −4, −1, 2, and 5 on the same set of coordinate axes. (a) f(x) = [[x]] + c (b) f(x) = [[x + c]]
Use the graph of y = f(x) to find the open intervals on which the graph of each transformation is increasing and decreasing. If not possible, state the reason.a. y = f(€“x)b. y =
1. Management originally predicted that the profits from the sales of a new product could be approximated by the graph of the function f shown. The actual profits are represented by the graph of the
For each function, sketch the graphs of the function when c = ˆ’3, ˆ’2, 1, and 2 on the same set of coordinate axes.
Use the graph of f to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com.a. y = f(-x)b. y = f(x) + 4c. y = 2f(x)d. y = - f(x - 4)e. y = f(x) - 3f. y = - f(x) - 1g. y =
1. Two functions f and g can be combined by the arithmetic operations of ________, ________, ________, and _________ to create new functions. 2. The ________ of the function f with the function g is
Evaluate the function for f(x) = x + 3 and g(x) = x2 - 2. 1. (f + g) (2) 2. (f + g) (−1) 3. (f − g) (0)
Use a graphing utility to graph f, g, and f + g in the same viewing window. Which function contributes most to the magnitude of the sum when 0 ≤ x ≤ 2? Which function contributes most to the
Find (a) f ∘ g, (b) g ∘ f, (c) g ∘ g. 1. f (x) = x + 8, g(x) = x − 3 2. f (x) = −4x, g(x) = x + 7
Use the graphs of f and g to graph h(x) = (f + g) (x). To print an enlarged copy of the graph, go to MathGraphs.com.1.2.
Find (a) f ∘ g (b) g ∘ f. Find the domain of each function and of each composite function. f (x) = √x + 4, g(x) = x2
On the same set of coordinate axes, (a) Graph the functions f, g, and f + g (b) Graph the functions f, g, and f ∘ g. 1. f (x) = 1/2x, g(x) = x − 4 2. f (x) = x + 3, g(x) = x2
Use the graphs of f and g to evaluate the functions.1. (a) (f + g) (3) (b) (f/g) (2) 2. (a) (f ˆ’ g) (1) (b) (fg) (4) 3. (a) (f ˆ˜ g) (2) (b) (g ˆ˜ f ) (2) 4. (a) ( f
Find two functions f and g such that (f ∘ g) (x) = h(x). (There are many correct answers.) 1. h(x) = (2x + 1)2 2. h(x) = (1 − x)3 3. h(x) = 3√x2 − 4
Find (a) (f + g) (x), (b) (f - g) (x), (c) (fg) (x), and (d) (f/g) (x). What is the domain f f/g? f (x) = x + 2, g(x) = x − 2
The research and development department of an automobile manufacturer determines that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the
The annual cost C (in thousands of dollars) and revenue R (in thousands of dollars) for a company each year from 2010 through 2016 can be approximated by the models C = 254 − 9t + 1.1t2 and R = 341
Let b(t) be the number of births in the United States in year t, and let d(t) represent the number of deaths in the United States in year t, where t = 10 corresponds to 2010. (a) If p(t) is the
Let d(t) be the number of dogs in the United States in year t, and let c(t) be the number of cats in the United States in year t, where t = 10 corresponds to 2010.(a) Find the function p(t) that
A square concrete foundation is a base for a cylindrical tank (see figure).(a) Write the radius r of the tank as a function of the length x of the sides of the square. (b) Write the area A of the
The number N of bacteria in a refrigerated food is given by N(T) = 10T2 − 20T + 600, 2 ≤ T ≤ 20 where T is the temperature of the food in degrees Celsius. When the food is removed from
You are a sales representative for a clothing manufacturer. You are paid an annual salary, plus a bonus of 3% of your sales over $500,000. Consider the two functions f(x) = x − 500,000 and g(x) =
The suggested retail price of a new hybrid car is p dollars. The dealership advertises a factory rebate of $2000 and a 10% discount. (a) Write a function R in terms of p giving the cost of the hybrid
1. If f (x) = x + 1 and g(x) = 6x, then (f ∘ g)(x) = (g ∘ f)(x). 2. When you are given two functions f and g and a constant c, you can find (f ∘ g) (c) if and only if g(c) is in the domain of
(a) Write a composite function that gives the oldest sibling's age in terms of the youngest. Explain how you arrived at your answer. (b) If the oldest sibling is 16 years old, find the ages of the
(a) Write a composite function that gives the youngest sibling's age in terms of the oldest. Explain how you arrived at your answer. (b) If the youngest sibling is 2 years old, find the ages of the
Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function.
Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.
Write two unique functions f and g such that (f ∘ g) (x) = (g ∘ f) (x) and f and g are (a) Linear functions (b) Polynomial functions with degrees greater than one.

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