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mathematics
precalculus
Questions and Answers of
Precalculus
Graph y = tan x and y = cot x together for -7 ≤ x ≤ 7. Comment on the behavior of cot x in relation to the signs and values of tan x.
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more
Use graphs to find approximate solution.3 - 2-x = 0
Use the addition formulas to derive the identities. COS X - TT 2 sin x
Gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.ƒ(x) =
Graph y = sin x and y = [sin x] together. What are the domain and range of [sin x]?
The population of Silver Run in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year.a. Estimate the population in 1915 and 1940.b. Approximately when did the
Use the addition formulas to derive the identities. cos x + TT 2 = -sin x
Gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.
Use the addition formulas to derive the identities. sin sin x + TT 2 = COS X
Find the (a) Domain and (b) Range. y = -x - 2, X, -x + 2, -2 ≤ x ≤-1 -1 < x < 1 1 < x < 2
Gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.ƒ(x) =
Graph the upper branch of the hyperbola y2 - 16x2 = 1.
Write a piecewise formula for the function. y 5 이 (2,5) 4 X
If Jean invests $2300 in a retirement account with a 6% interest rate compounded annually, how long will it take until Jean’s account has a balance of $4150?
Use the addition formulas to derive the identities. sin(x TT 2 = -cos X
Gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.ƒ(x) =
Graph two periods of the function ƒ(x) = 3cot x/2 + 1.
Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded continuously.
The table shows the average residential and transportation prices for energy consumption in the United States for the years 2000–2008, as reported by the U.S. Department of Energy. The prices are
Use the addition formulas to derive the identities.cos (A - B) = cos Acos B + sin Asin B
Gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x.ƒ(x) = x2
Graph the function ƒ(x) = sin3 x.
The federal minimum hourly wage rates have increased over the years. The table shows the rates at the year in which they first took effect, as reported by the U.S. Department of Labor.a. Make a
Use the addition formulas to derive the identities.sin (A - B) = sin Acos B - cos Asin B
The table shows the amount of yeast cells (measured as biomass) growing over a 7-hour period in a nutrient, as recorded by R. Pearl (1927) during a well-known biological experiment.a. Make a
Suppose that in any given year the number of cases of a disease is reduced by 20%. If there are 10,000 cases today, how many years will it takea. To reduce the number of cases to 1000?b. To eliminate
Express the given quantity in terms of sin x and cos x. COS З п 2 + x
What happens if you take B = A in the trigonometric identity cos (A - B) = cos Acos B + sin Asin B? Does the result agree with something you already know?
Show that the graph of the inverse of ƒ(x) = mx + b, where m and b are constants and m ≠ 0, is a line with slope 1/m and y-intercept -b/m.
What happens if you take B = 2p in the addition formulas? Do the results agree with something you already know?
Use the properties of logarithms to write the expression as a single term.a. ln sec θ + ln cos θ b. ln (8x + 4) - 2lncc. z/AuTદ 31n ²-1 - In (t + 1)
a. Find the inverse of ƒ(x) = -x + 1. Graph the line y = -x + 1 together with the line y = x. At what angle do the lines intersect?b. Find the inverse of ƒ(x) = -x + b (b constant). What angle does
Evaluate cos π/12. COS 117 12 as cos TT 2ㅠ + 3 4
Express the given quantity in terms of sin x and cos x.sin (2π - x)
Express the following logarithms in terms of ln 5 and ln 7.a. ln (1/125) b. ln 9.8c. ln 7√7 d. ln 1225e. ln 0.056 f. (ln 35 + ln (1/7))/(ln 25)
Describe how each graph is obtained from the graph of y = ƒ(x).a. y = ƒ(x - 5) b. y = ƒ(4x)c. y = ƒ(-3x) d. y = ƒ(2x + 1)e.f. y f = X 3 4
Evaluate sin 5π/12.
Find simpler expressions for the quantitie.a. eln (x2+y2) b. e-ln 0.3 c. elnpx-ln 2
Graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
Graph the functions.y = (x + 2)3/2 + 1
Find the function values.cos2 5π/12
Find simpler expressions for the quantitie.a. ln (esecθ) b. ln (e(ex)) c. ln (e2lnx)
Find the function values.sin2 3π/8
The standard formula for the tangent of the sum of two angles isDerive the formula. tan(A + B) tan Atan B 1 tan A tan B
Solve for y in terms of t or x, as appropriate.ln y = -t + 5
The accompanying figure shows the graph of a function g(t) with domain 3-4, 04 and range 3-3, 04. Find the domains and ranges of the following functions, and sketch their graphs.a. g(-t) b.
Apply the law of cosines to the triangle in the accompanying figure to derive the formula for cos (A - B). 1 y 1 A 0 В 1 X
Sketch the graph of the given function. What is the period of the function? = y COS TTX 2
Derive a formula for tan (A - B).Exercise 55The standard formula for the tangent of the sum of two angles isDerive the formula. tan(A + B) tan Atan B 1 tan A tan B
Graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
Solve for the angle θ, where 0 ≤ θ ≤ 2π.sin2 θ = cos2 θ
Solve for y in terms of t or x, as appropriate.ln (c - 2y) = t
Solve for the angle θ, where 0 ≤ θ ≤ 2π.cos 2θ + cos θ = 0
Solve for y in terms of t or x, as appropriate.ln (y2 - 1) - ln (y + 1) = ln (sin x)
Sketch the graph of the given function. What is the period of the function?y = sin(x/2)
Solve for k.a. e5k = 1/4b. 80ek = 1 c. e(ln 0.8)k = 0.8
Solve for t.a. e-0.01t = 1000b. ekt = 1/10c. e(ln 2)t = 1/2
a. Apply the formula for cos (A - B) to the identity sin θ = cos(π/2 - θ) to obtain the addition formula for sin (A + B).b. Derive the formula for cos (A + B) by substituting -B for B in the
Simplify the expression.a. 2log2 3 b. 10log10 (1/2) c. πlogπ 7d. log11 121 e. log121 11f. log3
The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, thenUse the accompanying figures and the identity sin(π - θ) = sin θ, if required, to derive
Solve for t.e(x2)e(2x+1) = et
A triangle has sides a = 2 and b = 3 and angle C = 40°. Find the length of side c.
A triangle has sides a = 2 and b = 3 and angle C = 60°. Find the sine of angle B using the law of sines.
Foridentify A, B, C, and D for the sine functions and sketch their graphs. f(x) = A sin ( ² 7 7 (x − c ) ) + - C) + D.
ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.a. Express a in terms of A and c.b. Express a in terms of A and b.
Express the ratios as ratios of natural logarithms and simplify.a.b.c. logg.x log3.x
It is often useful to know that, when x is measured in radians, sin x ≈ x for numerically small values of x. We will see why the approximation holds. The approximation error is less than 1 in 5000
Simplify the expression.a. 25log5 (3x2) b. loge(ex) c. log4(2ex sin x)
Foridentify A, B, C, and D for the sine functions and sketch their graphs. f(x) = A sin ( ² 7 7 (x − c ) ) + - C) + D.
ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.a. Express sin A in terms of a and c.b. Express sin A in terms of b and c.
You will explore graphically the general sine functionas you change the values of the constants A, B, C, and D. Use a CAS or computer grapher to perform the steps in the exercises.Set the constants A
You will explore graphically the general sine functionas you change the values of the constants A, B, C, and D. Use a CAS or computer grapher to perform the steps in the exercises.Set the constants A
Observers at positions A and B 2 km apart simultaneously measure the angle of elevation of a weather balloon to be 40° and 70°, respectively. If the balloon is directly above a point on the line
You will explore graphically the general sine functionas you change the values of the constants A, B, C, and D. Use a CAS or computer grapher to perform the steps in the exercises.Set the constants B
You will explore graphically the general sine functionas you change the values of the constants A, B, C, and D. Use a CAS or computer grapher to perform the steps in the exercises.Set the constants A
a. Graph the function ƒ(x) = sin x + cos(x/2).b. What appears to be the period of this function?c. Confirm your finding in part (b) algebraically.
a. Graph ƒ(x) = sin (1/x).b. What are the domain and range of ƒ?c. Is ƒ periodic? Give reasons for your answer.
If ƒ(x) is one-to-one and ƒ(x) is never zero, can anything be said about h(x) = 1/ƒ(x)? Is it also one-to-one? Give reasons for your answer.
If ƒ(x) = ln x and g(x) = 4 - x2, find the functions ƒ ∘ g, g ∘ ƒ, ƒ ∘ ƒ, g ∘ g, and their domains.
If a composite ƒ ∘ g is one-to-one, must g be one-to-one? Give reasons for your answer.
Use a graph to decide whether ƒ is one-to-one.a.b. 3 f(x) = x³ 1 X 2
Graph ln x, ln 2x, ln 4x, ln 8x, and ln 16x (as many as you can) together for 0 < x ≤ 10. What is going on? Explain.
Graph y = ln (x2 + c) for c = -4, -2, 0, 3, and 5. How does the graph change when c changes?
Graph y = ln |sin x | in the window 0 ≤ x ≤ 22, -2 ≤ y ≤ 0. Explain what you see. How could you change the formula to turn the arches upside down?
Graph the three functions y = xa, y = ax, and y = loga x together on the same screen for a = 2, 10, and 20. For large values of x, which of these functions has the largest values and which has the
Find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers and comment on any
Find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers and comment on any
Could xln 2 possibly be the same as 2ln x for x > 0? Graph the two functions and explain what you see.
Use a graph to find to 3 decimal places the values of x for which ex > 10,000,000.
a. Show that ƒ(x) = x3 and g(x) = 3√x are inverses of one another.b. Graph ƒ and g over an x-interval large enough to show the graphs intersecting at (1, 1) and (-1, -1). Be sure the picture
a. Show that h(x) = x3/4 and k(x) = (4x)1/3 are inverses of one another.b. Graph h and k over an x-interval large enough to show the graphs intersecting at (2, 2) and (-2, -2). Be sure the picture
The decay equation for radon-222 gas is known to be y = y0e-0.18t, with t in days. About how long will it take the radon in a sealed sample of air to fall to 90% of its original value?
Let P(5, −12) be a point on the circle x2 + y2 = 169 (see figure).(a) What is the slope of the line joining P and O(0, 0)?(b) Find an equation of the tangent line to the circle at P.(c) Let Q(x, y)
For positive numbers a < b, the pulse function is defined as (a) Sketch the graph of the pulse function.(b) Find the following limits:(i) lim x→a+ Pa,b(x)(ii) lim x→a− Pa,b(x)(iii) lim
Consider the function f(x) = √3 + x1 3 − 2 /x − 1.(a) Find the domain of f.(b) Use a graphing utility to graph the function.(c) Find lim x→−27+ f(x).(d) Find lim x→1 f(x).
Describe the Chain Rule for the composition of two differentiable functions in your own words.
List four notation alternatives to f′(x).
Describe the Quotient Rule in your own words.
Explain how to find the derivative of the function f(x) = cxn.
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