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mathematics
precalculus

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

Find the derivative of the function.y = ln |cos x / cos x - 1|
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point. f(x)= In 1+ sin²x, π In 2,
Find the area of the given region. Use a graphing utility to verify your result.y = csc(x + 1) -1 3- 1 1 2 X
Find the derivative of the function.y =ln|csc x|
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a). f(x) = sin x, sist π -SX: a 2
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a).f (x) = x3 + 3x - 1, a = -5
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent feature of a graphing utility to confirm your results. f(x) = 4x² - In(x + 1), (0,4)
Use implicit differentiation to find dy/dx.exy + x2 - y2 = 10
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a). π f(x) = cos 2x, 0≤x≤ ≤x≤ 2 a = 1
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a). f(x) = x + 3 x + l' x>-1, a = 2
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a).f (x) = 1/27(x5 + 2x3), a = - 11
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point.y = ln x4, (1, 0)
(a) Find the domains of f and f-1,(b) Find the ranges of f and f-1,(c) Graph f and f-1, and(d) Show that the slopes of the graphs of f and f-1 are reciprocals at the given points. Functions f(x) = x³ f-¹(x) = 3√x Point (1) (1/2)
Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point.xey + yex = 1, (0, 1)
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a). f(x) = x + 6 x - 2' x > 2, a = 3
Verify that f has an inverse function. Then use the function f and the given real number a to find (f-1)'(a).f (x) = √x - 4, a = 2
Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point.1 + ln xy = ex-y, (1, 1)
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point.f (x) = 3x2 - ln x, (1, 3)
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent feature of a graphing utility to confirm your results.y = ln x2/3, (-1, 0)
Find the second derivative of the function.g(x) = √x + ex ln x
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the pointy = x3 ln x4, (-1, 0)
Find the relative extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results.f (x) = (2 - x)ex
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent feature of a graphing utility to confirm your results.f (x) = sin 2x ln x2, (1, 0)
Find the average value of the function over the given interval.f(x) = 8/x2, [2, 4]
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. [ x [nl]] - dx = X In 2 In 1 In 2 = -
Find the average value of the function over the given interval. f(x) = 2 ln x X [1, e]
Find the average value of the function over the given interval. f(x) = sec π.Χ. 6 [0, 2]
(a) Find an equation of the tangent line to the graph of the function at the given point,(b) Use a graphing utility to graph the function and its tangent line at the point, and(c) Use the tangent feature of a graphing utility to confirm your results. 1 f(x) = -x In x², x In x², (-1,0)
Find the average value of the function over the given interval. f(x) = 4(x + 1) करे [2, 4]
Use the Midpoint Rule with n = 4 to approximate the value of the definite integral. Use a graphing utility to verify your result. 12 X - dx
Use the functions f (x) = 1/8x - 3 and g(x)= x3 to find the given value. (8)(₁-801-8)
Use logarithmic differentiation to find dy/dx. y = x(x - 1)³/2 √x+1 I < x
Use the functions f (x) = 1/8x - 3 and g(x)= x3 to find the given value. (f-¹ of ¹)(-2)
Use the Midpoint Rule with n = 4 to approximate the value of the definite integral. Use a graphing utility to verify your result. π/4 Jo sec x dx
Perform the following steps to find the maximum area of the rectangle shown in the figure(a) Solve for c in the equation f (c) = f (c + x).(b) Use the result in part (a) to write the area A as a function of x.(c) Use a graphing utility to graph the area function. Use the graph to approximate the
Use logarithmic differentiation to find dy/dx. y = (x + 1)(x − 2) (x − 1)(x + 2) - x > 2
A meteorologist measures the atmospheric pressure P (in millibars) at altitude h (in kilometers). The data are shown below.(a) Use a graphing utility to plot the points (h, ln P). Use the regression capabilities of the graphing utility to find a linear model for the revised data points.(b) The line
Is the functionconstant, increasing, or decreasing on the interval (0, ∞)?Explain. F(x) = 2x 1². = dt t
For 0 < x < y, use the Mean Value Theorem to show that 1 y V In y - ln x y - x L 1 X
Find a value of x such that 3 - dt = Cx 1 - dt. 1/4 t
A person walking along a dock drags a boat by a 10-meter rope. The boat travels along a path known as a tractrix (see figure). The equation of this path is(a) Use a graphing utility to graph the function.(b) What are the slopes of this path when x = 5 and x = 9?(c) What does the slope of the path
The table lists the approximate values V of a mid-sized sedan for the years 2010 through 2016. The variable t represents the time (in years), with t = 10 corresponding to 2010.(a) Use the regression capabilities of a graphing utility to fit linear and quadratic models to the data. Plot the data and
Find the area of the largest rectangle that can be inscribed under the curve y = e-x2 in the first and second quadrants.
Use the information in the graph of f below.(a) What is the slope of the tangent line to the graph of f-1 at the point (-1/2, -1)? Explain.(b) What is the slope of the tangent line to the graph of f-1 at the point (1, 2)? Explain. (-1,-1) 2 -3-2 y 3 2- -2- -34 f m=2, m = 2/1/2 (2, 1) + 1 2 3 Im X
Graph the functionon the interval [0, ∞).(a) Find the area bounded by the graph of f and the line y = 1/2x.(b) Determine the values of the slope m such that the line y = mx and the graph of f enclose a finite region.(c) Calculate the area of this region as a function of m. f(x) = X 1 + x²
Use the functions f(x) = x + 4 and g(x) = 2x - 5 to find the given function.(f º g)-1
Use the functions f(x) = x + 4 and g(x) = 2x - 5 to find the given function.(g º f)-1
Show thatis one-to-one and find (f-1)'(0). = 6: dt √2 √1 + 14 f(x) =
Use the graph of f' shown in the figure to answer the following.(a) Approximate the slope of f at x = -1. Explain.(b) Approximate any open intervals on which the graph of f is increasing and any open intervals on which it is decreasing. Explain. -5-4 3 2 1 - 1 y -2 -3+ 1 -X
Consider the function f(x) = xn, where n is odd. Does f-1 exist? Explain.
Show thatis one-to-one and find (f-1)'(0). (6x) = [₂² √² + F² d dt J2
The term t (in years) of a $200,000 home mortgage at 7.5% interest can be approximated bywhere x is the monthly payment in dollars.(a) Use a graphing utility to graph the model.(b) Use the model to approximate the term of a home mortgage for which the monthly payment is $1398.43. What is the total
The table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch).A model that approximates the data is T = 87.97 + 34.96 ln p + 7.91√p.(a) Use a graphing utility to plot the data and graph the model.(b) Find the rates of change of
Use implicit differentiation to find dy/dx.4xy + ln x2y = 7
Does adding a constant term to a function affect the existence of an inverse function? Explain.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. d [In(cx)] = [In x], -[In dx dx x], where c > 0
Prove that ∫ cot u du = ln|sin u|+ C.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Stª - X dx = In|cx|, c# 0
Let (a) Show that f is one-to-one if and only if bc - ad ≠ 0.(b) Given bc - ad ≠ 0, find f-1.(c) Determine the values of a, b, c, and d such that f = f-1. f(x) = = ax + b cx + d'
Let f and g be one-to-one functions. Prove that(a) f º g is one-to-one.(b) (f º g)-1(x) = (g-1 º f-1)(x).
Evaluate the definite integral. Use a graphing utility to verify your result. Sö e-2x dx
The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is one atmosphere (1.033227 kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers).(a) Use a graphing utility to find a model of
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.ln|x4|= ln x4
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.ln|cos θ2|= ln(cos θ2)
Evaluate the definite integral. Use a graphing utility to verify your result. Jo /2 xe-x/2 dx
Evaluate the definite integral. Use a graphing utility to verify your result. L₁ -1 el +4x dx
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.ln(an+m) = n ln a + m ln a, where a > 0 and m and n are rational.
Evaluate the definite integral. Use a graphing utility to verify your result. L xe xe x dx ax
Evaluate the definite integral. Use a graphing utility to verify your result. 1₂73 -2 ex+1 7- ex+1 dx
Evaluate the definite integral. Use a graphing utility to verify your result. ro J-2 x²ex²/2 dx
Evaluate the definite integral. Use a graphing utility to verify your result. J1 e3/x لاج dx
Suppose that f is a function on the interval [1, 3] such that -1 ≤ f (x) ≤ 1 for all x and ∫31 f (x) dx = 0. How large can ∫31 f (x)/x dx be?
Evaluate the definite integral. Use a graphing utility to verify your result. So Jo e4x 1 + 4x dx
Evaluate the definite integral. Use a graphing utility to verify your result. π/2 Jo esin ™x сos лx dx
Evaluate the definite integral. Use a graphing utility to verify your result. n/2 Jπ/3 esec 2x sec 2x tan 2x dx
Use the Midpoint Rule with n = 12 to approximate the value of the definite integral. Use a graphing utility to verify your result. [ √x ex dx
Find the general solution of the differential equation.dy/dx = xe9x2
Find the general solution of the differential equation.dy/dx = (ex - e-x)2
Find the particular solution of the differential equation that satisfies the initial conditions.f''(x) = sin x + e2x, f (0) = 1/4, f'(0) = 1/2
Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result.y = ex, y = 0, x = 0, x = 6
Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result.y = e-2x, y = 0, x = -1, x = 3
Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result.y = xe-x2/4, y = 0, x = 0, x = √6
Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result.y = e-2x + 2, y = 0, x = 0, x = 2
Use the Midpoint Rule with n = 12 to approximate the value of the definite integral. Use a graphing utility to verify your result. 2 Jo 2xe-x dx
A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given in the table.(a) Use the regression capabilities of a graphing utility to find a linear model for the points (t, ln R). Write the
Consider f (x) = xe-kx for k > 0. Find the relative extrema and the points of inflection of the function.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If f (x) = g(x)ex, then f'(x) = g'(x)ex.
Use a graphing utility to graph f (x) = ex and the given function in the same viewing window. How are the two graphs related?(a) g(x) = ex-2(b) h(x) = -1/2ex(c) q(x) = e-x + 3
Compare the asymptotes of the natural exponential function with those of the natural logarithmic function.
Use the result of Exercise 21 to show that the following functions are differentiable at x = 0.Data from in Exercise 21Suppose that the functions ƒ and g are defined throughout an open interval containing the point x0, that ƒ is differentiable at x0, that ƒ(x0) = 0, and that g is continuous at
Find the derivative of y with respect to the appropriate variable.y = sec-1 5s
Outline a strategy for solving related rates problems. Illustrate with an example.
What can be said about the extreme values of a function that is continuous on a closed interval?
Use Newton’s method to estimate the solutions of the equation x2 + x - 1 = 0. Start with x0 = -1 for the left-hand solution and with x0 = 1 for the solution on the right. Then, in each case, find x2.
Does ƒ(x) = x3 + 2x + tan x have any local maximum or minimum values? Give reasons for your answer.
What can you say about a function whose maximum and minimum values on an interval are equal? Give reasons for your answer.
Find values of a and b such that the functionhas a local extreme value of 1 at x = 3. Is this extreme value a local maximum, or a local minimum? Give reasons for your answer. f(x): ax + b 2 x²² - 1
Use the sign pattern for the derivativeto identify the points where ƒ has local maximum and minimum values. df dx 6(x - 1)(x - 2)²(x - 3)³(x-4)4
What does it mean for a function to have a local extreme value on its domain? An absolute extreme value? How are local and absolute extreme values related, if at all? Give examples.
Use Newton’s method to estimate the one real solution of x3 + 3x + 1 = 0. Start with x0 = 0 and then find x2.

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