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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Graph each function with a graphing utility using the given window. Then state the domain and range of the function.g(x) = (x2 - 4) √x + 5; [-5, 5] × [-10, 50]
Graph each function with a graphing utility using the given window. Then state the domain and range of the function.h(u) = 3√u - 1; [-7, 9] × [-2, 2]
Graph each function with a graphing utility using the given window. Then state the domain and range of the function.F(w) = 4√2 - w; [-3, 2] × [0, 2]
Graph each function with a graphing utility using the given window. Then state the domain and range of the function.f(x) = √4 - x2; [-4, 4] × [-4, 4]
Graph each function with a graphing utility using the given window. Then state the domain and range of the function. _y + 1 g(y) [-4, 6] × [-3, 3] (y + 2)(y – 3)'
Graph each function with a graphing utility using the given window. Then state the domain and range of the function.f(x) = 3x4 - 10; [-2, 2] × [-10, 15]
Decide whether graphs A, B, or both represent functions. УА х
Decide whether graphs A, B, or both represent functions. y. х
Sketch a graph of an odd function f and state how f(x) and f(-x) are related.
Sketch a graph of an even function f and state how f(x) and f(-x) are related.
Explain how to find the domain of f ͦ g if you know the domain and range of f and g.
Suppose f and g are even functions with f(2) = 2 and g(2) = -2. Evaluate f(g(2)) and g(f(-2)).
If f (x) = √x and g(x) = x3 – 2, find the compositions f ͦ g, g ͦ f, f ͦ f, and g ͦ g.
Which statement about a function is true? (i) For each value of x in the domain, there corresponds one unique value of y in the range; (ii) for each value of y in the range, there corresponds one
If f(x) = 1/(x3 + 1), what is f (2)? What is f(y2)?
Explain how the vertical line test is used to detect functions.
Is the independent variable of a function associated with the domain or range? Is the dependent variable associated with the domain or range?
Use the terms domain, range, independent variable, and dependent variable to explain how a function relates one variable to another variable.
Graph the parabola f(x) = x2. Explain why the secant lines between the points (-a, f(-a)) and (a, f(a)) have zero slope. What is the slope of the tangent line at x = 0?
Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph.
Describe a process for finding the slope of the line tangent to the graph of f at (a, f(a)).
What is the slope of the secant line that passes through the points (a, f(a)) and (b, f(b)) on the graph of f?
Suppose s(t) is the position of an object moving along a line at time t ≥ 0. Describe a process for finding the instantaneous velocity at t = a.
Suppose s(t) is the position of an object moving along a line at time t ≥ 0. What is the average velocity between the times t = a and t = b?
A common way of displaying a sphere (such as Earth) on a plane (such as a map) is to use a stereo-graphic projection. Here is the two-dimensional version of the method, which maps a circle to a line.
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.cos (2 sin-1 x)
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.sin (2 cos-1 x)
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.sin-1 x + sin-1 (-x)
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.csc-1 (sec θ)
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.cot-1 (tan θ)
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.tan (sec-1 (x/2))
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.sin (cos-1 (x/2))
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ u ≤ π/2.cos (tan-1 x)
Given that θ = sin-1 12/13, evaluate cos θ, tan θ, cot θ, sec θ, and csc θ.
Evaluate or simplify the following expressions without using a calculator.cos-1 (sin 3π)
Evaluate or simplify the following expressions without using a calculator.sin (sin-1 x)
Evaluate or simplify the following expressions without using a calculator.cos (cos-1 (-1))
Evaluate or simplify the following expressions without using a calculator.sin-1 (-1)
Evaluate or simplify the following expressions without using a calculator.cos-1 (-1/2)
Evaluate or simplify the following expressions without using a calculator.cos-1 √3/2
Evaluate or simplify the following expressions without using a calculator.sin-1 √3/2
Find the points at which the curves intersect on the given interval.y = sin x and y = -1/2 on (0, 2π)
Find the points at which the curves intersect on the given interval.y = sec x and y = 2 on (-π/2, π/2)
Match each function a-f with the corresponding graphs A–F.a. f(x) = -sin xb. f(x) = cos 2xc. f(x) = tan (x/2)d. f(x) = -sec xe. f(x) = cot 2xf. f(x) = sin2 x (A) (B) Na- (C)
Find a trigonometric function f represented by the graph in the figure. УА
Find a trigonometric function f that satisfies each set of properties. Answers are not unique.a. It has a period of 6 with a minimum value of -2 at t = 0 and a maximum value of 2 at t = 3.b. It has a
Use shifts and scalings to graph the following functions, and identify the amplitude and period.a. f(x) = 4 cos (x/2)b. g(θ) = 2 sin (2πθ/3)c. h(θ) = -cos(2(θ - π/4))
a. Convert 135° to radian measure.b. Convert 4π/5 to degree measure.c. What is the length of the arc on a circle of radius 10 associated with an angle of 4π/3 (radians)?
Find the inverse on the specified interval and express it in the form y = f-1(x). Then graph f and f-1.f(x) = 1/x2, for x > 0
Find the inverse on the specified interval and express it in the form y = f-1(x). Then graph f and f-1.f(x) = x2 - 4x + 5, for x > 2
Determine the intervals on which the following functions have an inverse.g(t) = 2 sin (t/3)
Determine the intervals on which the following functions have an inverse.f(x) = x3 - 3x2
The figure shows the graphs of y = 2x, y = 3-x, and y = - ln x. Match each curve with the correct function. У х
Solve the equation log10 x2 + 3 log10 x = log10 32 for x. Does the answer depend on the base of the log in the equation?Use properties of logarithms and exponentials, not a calculator, for the
Solve the equation 48 = 6e4k for k.Use properties of logarithms and exponentials, not a calculator, for the following exercises.
Identify the symmetry (if any) in the graphs of the following equations.a. y = cos 3xb. y = 3x4 - 3x2 + 1c. y2 - 4x2 = 4
Evaluate and simplify the difference quotients f(x + h) - f(x)/h and f(x) - f(a)/x - a for each function.f(x) = 7/x + 3
Evaluate and simplify the difference quotients f(x + h) - f(x)/h and f(x) - f(a)/x - a for each function.f(x) = x3 + 2
Evaluate and simplify the difference quotients f(x + h) - f(x)/h and f(x) - f(a)/x - a for each function.f(x) = 4 - 5x
Evaluate and simplify the difference quotients f(x + h) - f(x)/h and f(x) - f(a)/x - a for each function.f(x) = x2 - 2x
Find functions f and g such that h = f ◦ g.a. h(x) = sin(x2 + 1)b. h(x) = (x2 - 4)-3c. h(x) = ecos 2x
Let f(x) = x3, g(x) = sin x, and h(x) = √x.a. Evaluate h(g(π/2)).b. Find h(f(x)).c. Find f(g(h(x2)).d. Find the domain of g ◦ f.e. Find the range of f ◦ g.
The graph of f is shown in the figure. Graph the following functions.a. f(x + 1)b. 2 f(x - 1)c. -f(x/2)d. f(2(x - 1)) y = f(x) х
Starting with the graph of f(x) = x2, plot the following functions. Use a graphing calculator to check your work.a. f(x + 3)b. 2f(x - 4)c. -f(3x)d. f(2(x - 3))
A small publisher plans to spend $1000 for advertising a paperback book and estimates the printing cost is $2.50 per book. The publisher will receive $7 for each book sold.a. Find the function C =
Water boils at 212° F at sea level and at 200° F at an elevation of 6000 ft. Assume that the boiling point B varies linearly with altitude a. Find the function B = f(a) that describes the
Graph the equations y = x2 and x2 + y2 - 7y + 8 = 0. At what point(s) do the curves intersect?
Find the domain and range of the functions f(x) = x1/7 and g(x) = x1/4.
Graph the functions f(x) = x1/3 and g(x) = x1/4. Find all points where the two graphs intersect. For x > 1, is f(x) > g(x) or is g(x) > f(x)?
Graph the following equations. Use a graphing utility to check your work.a. 2x - 3y + 10 = 0b. y = x2 + 2x - 3c. x2 + 2x + y2 + 4y + 1 = 0d. x2 - 2x + y2 - 8y + 5 = 0
Suppose you plan to take a 500-mile trip in a car that gets 35 mi/gal. Find the function C = f(p) that gives the cost of gasoline for the trip when gasoline costs $p per gallon.
Consider the function f(x) = 2(x - |x|). Express the function in two pieces without using the absolute value. Then graph the function by hand. Use a graphing utility to check your work.
The parking costs in a city garage are $2 for the first half hour and $1 for each additional half hour. Graph the function C = f(t) that gives the cost of parking for t hours, where 0 ≤ t ≤ 3.
In each part below, find an equation of the line with the given properties. Graph the line.a. The line passing through the points (2, -3) and (4, 2)b. The line with slope 3/4 and x-intercept (-4,
Find the domain and range of the following functions.a. f(x) = x5 + √xb. g(y) = 1/y - 2c. h(z) = √z2 - 2z - 3
Determine whether the following statements are true and give an explanation or counterexample.a. A function could have the property that f(-x) = f(x), for all x.b. cos (a + b) = cos a + cos b, for
Use the figure to prove the law of sines: sin A/a = sin B/b = sin C/c. a
Use the figure to prove the law of cosines (which is a generalization of the Pythagorean theorem): c2 = a2 + b2 - 2ab cos θ. УА (b cos 0, b sin 0) b' a (a, 0) х
Prove that the area of a sector of a circle of radius r associated with a central angle θ (measured in radians) is A = 1/2 r2 θ.
An auditorium with a flat floor has a large flatpanel television on one wall. The lower edge of the television is 3 ft above the floor, and the upper edge is 10 ft above the floor (see figure).
The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the
A pole of length L is carried horizontally around a corner where a 3-ft-wide hallway meets a 4-ft-wide hallway. For 0 < θ < π/2, find the relationship between L and u at the moment when the
Two ladders of length a lean against opposite walls of an alley with their feet touching (see figure). One ladder extends h feet up the wall and makes a 75° angle with the ground. The other ladder
A boat approaches a 50-ft-high lighthouse whose base is at sea level. Let d be the distance between the boat and the base of the lighthouse. Let L be the distance between the boat and the top of the
A light block hangs at rest from the end of a spring when it is pulled down 10 cm and released. Assume the block oscillates with an amplitude of 10 cm on either side of its rest position with a
Verify that the function has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.a. It has a period of 365 days.b. Its maximum and minimum
The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles: one by wrapping a rope around the
Near the end of the 1950 Rose Bowl football game between the University of California and Ohio State University, Ohio State was preparing to attempt a field goal from a distance of 23 yd from the
It has a period of 24 hr with a minimum value of 10 at t = 3 hr and a maximum value of 16 at t = 15 hr.Design a sine function with the given properties.
It has a period of 12 hr with a minimum value of -4 at t = 0 hr and a maximum value of 4 at t = 6 hr.Design a sine function with the given properties.
Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.q(x) = 3.6 cos
Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.p(x) = 3 sin
Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.g(x) = -2 cos
Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.f(x) = 3 sin 2x
Identify the amplitude and period of the following functions.q(x) = 3.6 cos (πx/24)
Identify the amplitude and period of the following functions.p(t) = 2.5 sin (12/(t - 3))
Identify the amplitude and period of the following functions.g(θ) = 3 cos (θ/3)
Identify the amplitude and period of the following functions.f(θ) = 2 sin 2θ
Given the following information about one trigonometric function, evaluate the other five functions.csc θ = 13/12 and 0 < θ < π/2
Given the following information about one trigonometric function, evaluate the other five functions.sec θ = 5/3 and 3π/2 < θ < 2π

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