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mathematics
precalculus

Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions

Sketch the graphs of the rational function y || = x + 1 x - 3
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing calculator or computer grapher. k(x) = x³ + 3x² + 3x + 1, ·∞ < x≤ 0
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. [(ex (ex + 4³) dx
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing calculator or computer grapher. f(x) = √25 – x², −5 ≤ x ≤ 5
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = ex - 2e-x - 3x
Graph each function. Then use the function’s first derivative to explain what you see.y = x2/3 - (x - 1)1/3
Sketch the graphs of the rational function y = 2x x + 5
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. fa. (1.3)* dx
A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = 2x2 - 8x + 9
Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea or nautical miles per hour).
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing calculator or computer grapher. f(x) = √x² – 2x − 3, 3 ≤ x
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = xe-x
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = x3 - 2x + 4
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing calculator or computer grapher. g(x) = x - 2 x² - 1' 0 ≤ x ≤ 1
Sketch the graphs of the rational function. y || x² = x + 1 X
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. S (4 sec x tan x - 2 sec² x) dx
A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = ln (cos x)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = In x Vx
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing calculator or computer grapher. g(x) = x² 4 - x² -2 < x≤ 1
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. (csc² x - csc x cot x) dx
Sketch the graphs of the rational function. y || x³ + 2 2x
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = x3 + x2 - 8x + 5
Show that at some instant during a 2-hour automobile trip the car’s speedometer reading will equal the average speed for the trip.
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = 1 1 + ex
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = x3(x - 5)2
On our moon, the acceleration of gravity is 1.6 m/sec2. If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?
The geometric mean of two positive numbers a and b is the number √ab. Show that the value of c in the conclusion of the Mean Value Theorem for ƒ(x) = 1/x on an interval of positive numbers [a, b] is c = √ab.
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. y = √x² - 1
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. (sin 2x-csc² x) dx
Sketch the graphs of the rational function y = x4 1 x² 2
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) = sin 2x, sin 2x, 0≤ x ≤ T
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y || et 1 + et
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. [120 (2 cos 2x - 3 sin 3x) dx
Sketch the graphs of the rational function. y || x² 4 3
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) = sin x - COS X, 0≤ x ≤ 2T
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) = V3 cos x + sin x, 0≤x≤ 2T
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. + cos 4t -dt 2
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) = -2x + tan x, F|N < x < 크 TT 2
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. y 1 1x2
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = x - 4√x
Sketch the graphs of the rational function y = x² er 4
The arithmetic mean of two numbers a and b is the number (a + b)/2. Show that the value of c in the conclusion of the Mean Value Theorem for ƒ(x) = x2 on any interval [a, b] is c = (a + b)/2.
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. y = √3 + 2x = x² V3 -
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. - cos 6t dt 2 1-
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) -2-2 sin 2, 2' = 0 ≤ x ≤ 2TT
Graph the function ƒ(x) = sin x sin (x + 2) - sin2 (x + 1). What does the graph do? Why does the function behave this way? Give reasons for your answers.
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. y = X x² + 1 20
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. cos x -2 cos x - cos² x, f(x) = −2 -T≤ x ≤ T
a. Construct a polynomial ƒ(x) that has zeros at x = -2, -1, 0, 1, and 2.b. Graph ƒ and its derivative ƒ′ together. How is what you see related to Rolle’s Theorem?c. Do g(x) = sin x and its derivative g′ illustrate the same phenomenon as ƒ and ƒ′?
Assume that ƒ is continuous on [a, b] and differentiable on (a, b). Also assume that ƒ(a) and ƒ(b) have opposite signs and that ƒ′ ≠ 0 between a and b. Show that ƒ(x) = 0 exactly once between a and b.
Sketch the graphs of the rational function. y x² + 1 2 X
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. y = x + 1 x² + 2x + 2
Discuss the extreme-value behavior of the function ƒ(x) = x sin (1/x), x ≠ 0. How many critical points does this function have? Where are they located on the x-axis? Does ƒ have an absolute minimum? An absolute maximum?
If the graphs of two differentiable functions ƒ(x) and g(x) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) = csc² x csc² x 2 cot x, 0 < x < T
a. Find the local extrema of each function on the given interval, and say where they occur.b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′. f(x) = sec² x 2 tan x, -T 2
Sketch the graph of a differentiable function y = ƒ(x) through the point (1, 1) if ƒ′(1) = 0 anda. ƒ′(x) > 0 for x < 1 and ƒ′(x) < 0 for x > 1;b. ƒ′(x) < 0 for x < 1 and ƒ′(x) > 0 for x > 1;c. ƒ′(x) > 0 for x ≠ 1;d. ƒ′(x) < 0 for x ≠ 1.
Show that the function have local extreme values at the given values of θ, and say which kind of local extreme the function has. h(0) = 0 5 sin 2' 0 ≤ 0 ≤ π, at 0 = 0 and 0 = t TT
Show that the function have local extreme values at the given values of θ, and say which kind of local extreme the function has. h(0) = 3 cos 0 2² 0 ≤ 0 ≤ 2π, at 0 = 0 and 0 = 27
Sketch the graph of a continuous function y = g(x) such thata.b. g(2) = 2,0 < g' < 1 for x < 2, g'(x) → -1 < g' < 0 for x > 2, and g'(x)→-1 1¯ as x→2¯, as x→ 2+;
Assume that ƒ and g are differentiable on [a, b] and that ƒ(a) = g(a) and ƒ(b) = g(b). Show that there is at least one point between a and b where the tangents to the graphs of ƒ and g are parallel or the same line. Illustrate with a sketch.
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = ex + e-x
L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. lim √9x + 1 √x + 1
Suppose that ƒ′(x) ≤ 1 for 1 ≤ x ≤ 4. Show that ƒ(4) - ƒ(1) ≤ 3.
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = ex - e-x
Suppose that 0 < ƒ′(x) < 1>2 for all x-values. Show that ƒ(-1) < ƒ(1) < 2 + ƒ(-1).
Show that |cos x - 1| ≤ |x| for all x-values.
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = x ln x
Show that for any numbers a and b, the sine inequality |sin b - sin a| ≤ |b - a| is true.
L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. Vx lim x0+ √sinx
Sketch the graph of a continuous function y = h(x) such thata.b. h(0) = 0,−2 ≤ h(x) ≤ 2 for all x, h'(x) →∞as x→0¯¯, and h'(x)→→∞ as x→ 0¹;
Let ƒ be a function defined on an interval [a, b]. What conditions could you place on ƒ to guarantee thatwhere min ƒ′ and max ƒ′ refer to the minimum and maximum values of ƒ′ on [a, b]? Give reasons for your answers. min f' ≤ f(b) f(a) b - a < max f',
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = x2 ln x
L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. cot x lim x0+ CSC X
L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. sec x lim x→(π/2)- tan x
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = cos-1 (x2)
If |ƒ(w) - ƒ(x)| ≤ |w - x| for all values w and x and ƒ is a differentiable function, show that -1 ≤ ƒ′(x) ≤ 1 for all x-values.
L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. 2.x 3x lim xxx 3x + 4x
Sketch the graph of a differentiable function y = ƒ(x) that hasa. A local minimum at (1, 1) and a local maximum at (3, 3);b. A local maximum at (1, 1) and a local minimum at (3, 3);c. Local maxima at (1, 1) and (3, 3);d. Local minima at (1, 1) and (3, 3).
Use the inequalities in Exercise 70 to estimate ƒ(0.1) if ƒ′(x) = 1/(1 + x4 cos x) for 0 ≤ x ≤ 0.1 and ƒ(0) = 1.Exercise 70Let ƒ be a function defined on an interval [a, b]. What conditions could you place on ƒ to guarantee thatwhere min ƒ′ and max ƒ′ refer to the minimum and
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.y = sin-1(ex)
Assume that ƒ is differentiable on a ≤ x ≤ b and that ƒ(b) < ƒ(a). Show that ƒ′ is negative at some point between a and b.
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. y xV4 – x²
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.y = x2/3(x + 2)
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.y = x2/3(x2 - 4)
L’Hôpital’s Rule does not help with the limit. Try it—you just keep on cycling. Find the limits some other way. 2x + 4* lim x-x5x - 2x
Use the inequalities in Exercise 70 to estimate ƒ(0.1) if ƒ′(x) = 1/(1 - x4) for 0 ≤ x ≤ 0.1 and ƒ(0) = 2.Exercise 70Let ƒ be a function defined on an interval [a, b]. What conditions could you place on ƒ to guarantee thatwhere min ƒ′ and max ƒ′ refer to the minimum and maximum
Find the open intervals on which the function ƒ(x) = ax2 + bx + c, a ≠ 0, is increasing and decreasing. Describe the reasoning behind your answer.
Shows the graphs of the first and second derivatives of a function y = ƒ(x). Copy the picture and add to it a sketch of the approximate graph of ƒ, given that the graph passes through the point P. y P 0 y = f'(x) A y=f"(x) X
Right, or wrong? Say which for each formula and give a brief reason for each answer.a.b.c. [xs x sin x dx 2²₁ 2 sin x + C
Verify the formula differentiation. tan x x² dx = In x − In (1 + x²) tan ¹x X + C
Use the results of Exercise 81 to show that the function have inverses over their domains. Find a formula for dƒ -1/dx using Theorem 3ƒ(x) = (1/3)x + (5/6)Exercise 81Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x1 and x2 in I, x2 ≠ x1
Use the results of Exercise 81 to show that the function have inverses over their domains. Find a formula for dƒ -1/dx using Theorem 3ƒ(x) = 27x3Exercise 81Show that increasing functions and decreasing functions are one to- one. That is, show that for any x1 and x2 in I, x2 ≠ x1 implies ƒ(x2)
Right, or wrong? Say which for each formula and give a brief reason for each answer.a.b.c. Ita tan Ꮎ sec- Ꮎ dᎾ sec³ 0 3 + C
Use the results of Exercise 81 to show that the function have inverses over their domains. Find a formula for dƒ -1/dx using Theorem 3ƒ(x) = 1 - 8x3Exercise 81Show that increasing functions and decreasing functions are one to- one. That is, show that for any x1 and x2 in I, x2 ≠ x1 implies
Shows the graphs of the first and second derivatives of a function y = ƒ(x). Copy the picture and add to it a sketch of the approximate graph of ƒ, given that the graph passes through the point P. 0 P y = f'(x) A y=f"(x) X
Right, or wrong? Say which for each formula and give a brief reason for each answer.a.b.c. √(2x. (2x + 1)² dx = (2x + 1)³ 3 + C
Use the results of Exercise 81 to show that the function have inverses over their domains. Find a formula for dƒ -1/dx using Theorem 3.ƒ(x) = (1 - x)3Exercise 81Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x1 and x2 in I, x2 ≠ x1 implies

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