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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

Determine whether there exist any values of x in the interval [0, 2π) such that the rate of change of ƒ(x) = sec x and the rate of change of g(x) =csc X areequal.
In Exercises find the second derivative of the function.ƒ(x) = sin x²
In Exercises find the second derivative of the function.ƒ(x) = sec ² πx
In Exercises evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. f(x) = 1 √x + 4' 0,
In Exercises find the second derivative of the function. f(x) = 4x52x³ + 5x²
In Exercises find the second derivative of the function. f(x) = 4x³/2
In Exercises evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result.ƒ(x) = cos x², (0, 1)
In Exercises evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. g(t) = tan 2t, √√√3
In Exercises find the second derivative of the function. f(x) = = x-1
In Exercises find the second derivative of the function. f(x) = x² + 3x-3
In Exercises find the second derivative of the function. f(x) x2 + 3x X x - 4
In Exercises the graphs of a functionƒ and its derivative ƒ' are shown. Label the graphs as ƒ or ƒ' and write a short paragraph stating the criteria you used in making your selection. -2 32 -2 -3 y 3 X
In Exercises the graphs of a functionƒ and its derivative ƒ' are shown. Label the graphs as ƒ or ƒ' and write a short paragraph stating the criteria you used in making your selection. y 3 www. 3 X
In Exercises find the second derivative of the function. f(x) = x sin x
In Exercises the graphs of a functionƒ and its derivative ƒ' are shown. Label the graphs as ƒ or ƒ' and write a short paragraph stating the criteria you used in making your selection. ++ 432 2 y < H 1 2 3 4 X
In Exercises the graphs of a functionƒ and its derivative ƒ' are shown. Label the graphs as ƒ or ƒ' and write a short paragraph stating the criteria you used in making your selection. < + 432 X 4 34 IT
In Exercises find the second derivative of the function. f(x) = sec X x
In Exercises find the given higher-order derivative. (x) (+) £ *x^2 = (x)mu£
The table shows some values of the derivative of an unknown function ƒ. Complete the table by finding the derivative of each transformation of ƒ, if possible.(a) g(x) = ƒ(x) - 2(b) h(x) = 2ƒ(x)(c) r(x) = ƒ(-3x)(d) s(x) = ƒ(x + 2) X f'(x) g'(x) h'(x) r'(x) s'(x) -2 -1 4 를 0 1 - 1 2 -2 3 - 4
In Exercises find the given higher-order derivative. f'(x) = x², f'(x)
In Exercises find the given higher-order derivative. 2 ƒ"(x) = 2 — ²/² f(x) X
Given that g(5) = -3, g'(5) = 6, h(5)= 3, and h'(5) = -2, find ƒ'(5) for each of the following, if possible. If it is not possible, state what additional information is required.(a)(b)(c)(d) f(x) = g(x)h(x)
In Exercises find the given higher-order derivative. f(4)(x) = 2x + 1, f)(x)
In Exercises use the given information to find ƒ'(2).ƒ(x) = 2g(x) + h(x) g(2) = 3 and g(2) = -2 h(2) 1 and h'(2) = 4 =
In Exercises the graphs of ƒ and g are shown. Let h(x) = ƒ(g(x)) and s(x) = g(ƒ(x)). Find each derivative, if it exists. If the derivative does not exist, explain why.(a) Find h'(1). (b) Find s'(5). y +10 ++ ----8- A H 246810. -X
In Exercises use the given information to find ƒ'(2).ƒ(x) = 4 - h(x) g(2) = 3 and g(2) = -2 h(2) 1 and h'(2) = 4 =
In Exercises the graphs of ƒ and g are shown. Let h(x) = ƒ(g(x)) and s(x) = g(ƒ(x)). Find each derivative, if it exists. If the derivative does not exist, explain why.(a) Find h'(3).(b) Find s'(9). 10- H y 2 4 6 8 10 X
In Exercises use the given information to find ƒ'(2).ƒ(x) = g(x)h(x) g(2) = 3 and g(2) = -2 h(2) 1 and h'(2) = 4 =
In Exercises the graphs of ƒ, ƒ', and ƒ" are shown on the same set of coordinate axes.Identify each graph. Explain your reasoning. To print anenlarged copy of the graph. -2 ㅜ 2 y + 2 X
The displacement from equilibrium of an object in harmonic motion on the end of a spring iswhere y is measured in feet and t is the time in seconds. Determine the position and velocity of the object when t = π/8. 1 ==co cos 12t - 1 4 sin 12t
In Exercises the graph of ƒ is shown. Sketch the graphs of ƒ' and ƒ". To print an enlargedcopy of the graph. -4/-2 y f 4 2 -2 + 4
In Exercises use the given information to find ƒ'(2). g(2) = 3 and g(2) = -2 h(2) 1 and h'(2) = 4 =
In Exercises the graphs of ƒ, ƒ', and ƒ" are shown on the same set of coordinate axes. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph. y # -1 3 -1 -27 + X نرا
Sketch the graph of a differentiable function ƒ such that ƒ > 0 and ƒ' < 0 for all real numbers x. Explain how you found your answer.
Sketch the graph of a differentiable function ƒ such that ƒ(2) = 0, ƒ' < 0 for -∞ < x < 2, and ƒ' > 0 for 2 < x < ∞. Explain how you found your answer.
A 15-centimeter pendulum moves according to the equation θ = 0.2 cos 8t, where θ is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of θ when t = 3 seconds.
A buoy oscillates in simple harmonic motion y = A cos ωt as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its
The number N of bacteria in a culture after days is modeled byFind the rate of change of N with respect to when(a) t = 0,(b) t = 1,(c) t = 2, (d) t = 3, (e) t = 4,(f) What can you conclude? 3 400[1 (t² + 2)²] N = 400 1
In Exercises the graph of ƒ is shown. Sketch the graphs of ƒ' and ƒ". To print an enlarged copy of the graph. -8 f 8 4 -4. 4 ➤X
In Exercises the graph of ƒ is shown. Sketch the graphs of ƒ' and ƒ". To print an enlarged copy of the graph. -4 RIN X
The velocity of an automobile starting from rest iswhere is measured in feet per second. Find the acceleration at (a) 5 seconds(b) 10 seconds(c) 20 seconds v(t) = 100t 2t + 15
In Exercises the graph of ƒ is shown. Sketch the graphs of ƒ' and ƒ". To print an enlarged copy of the graph. 2 1 y -1 -2 + ка П Зл 2 2п X
The value V of a machine t years after it is purchased is inversely proportional to the square root of t + 1. The initial value of the machine is $10,000.(a) Write V as a function of t.(b) Find the rate of depreciation when t = 1.(c) Find the rate of depreciation when t = 3.
Let ƒ be a differentiable function of period p.(a) Is the function ƒ' periodic? Verify your answer.(b) Consider the function g(x) = ƒ(2x). Is the function g'(x)periodic? Verify your answer.
A car is traveling at a rate of 66 feet per second (45 miles per hour) when the brakes are applied. The position function for the car is s(t) = -8.25t² + 66t, where s is measured in feet and t is measured in seconds. Use this function to complete the table, and find the average velocity during
The figure shows the graphs of the position, velocity, and acceleration functions of a particle.(a) Copy the graphs of the functions shown. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph, go to MathGraphs.com.(b) On your sketch, identify when the particle speeds
Let u be a differentiable function of x. Use the fact that |u| = √u² to prove that d dx [디ul] = wi n 이미 u≠ 0.
The velocity of an object in meters per second is v(t) = 36 - 1² for 0 ≤ t ≤ 6. Find the velocity and acceleration of the objectwhen t = 3. What can be said about the speed of the objectwhen the velocity and acceleration have opposite signs?
In Exercises develop a general rule for ƒ(n)(x) given ƒ(x). X = (x) f
(a) Find the derivative of the function g(x) = sin² x + cos²x in two ways.(b) For ƒ(x) = sec² x and g(x) = tan² x, show that ƒ'(x) = g'(x).
In Exercises find the derivatives of the function ƒ for n = 1, 2, 3, and 4. Use the results to write a general rule for ƒ'(x) in terms of n. f(x) = x" sin x
In Exercises develop a general rule for ƒ(n)(x) given ƒ(x). f(x) = xn
Where are the functions ƒ1(x) = |sin x| and ƒ2(x) = sin |x| differentiable?
In Exercises find the derivatives of the function ƒ for n = 1, 2, 3, and 4. Use the results to write a general rule for ƒ'(x) in terms of n. f(x) = COS X xn
In Exercises (a) Find the specified linear and quadratic approximations of ƒ, (b) Use a graphing utility to graph ƒ and the approximations, (c) Determine whether P1 or P2 is the better approximation (d) State how the accuracy changes as you move farther from x = a. f(x) = sec
(a) Show that the derivative of an odd function is even. That is, if ƒ(-x) = -ƒ(x), then ƒ'(-x) = ƒ'(x).(b) Show that the derivative of an even function is odd. That is, if ƒ(-x) = ƒ(x), then ƒ'(-x) = -ƒ'(x).
In Exercises (a) Find the specified linear and quadratic approximations of ƒ, (b) Use a graphing utility to graph ƒ and the approximations, (c) Determine whether P1 or P2 is the better approximation (d) State how the accuracy changes as you move farther from x = a. f(x) = tan
In Exercises verify that the function satisfies the differential equation. Function y = 2x³ - 6x + 10 Differential Equation -y" - xy" - 2y = -24x²
In Exercises verify that the function satisfies the differential equation. Function y = 2 sin x + 3 Differential Equation y"+y = 3
Consider the function ƒ(x) = g(x)h(x).(a) Use the Product Rule to generate rules for finding ƒ"(x), ƒ'''(x), and ƒ(4)(x).(b) Use the results of part (a) to write a general rule for ƒ(n)(x).
In Exercises verify that the function satisfies the differential equation. Function 1 y X , x > 0 Differential Equation x³ y" + 2x²y = 0
Develop a general rule for [xƒ(x)](n), where f is a differentiable function of x.
Let k be a fixed positive integer. The nth derivative of has the form where Pn(x) is a polynomial. Find Pn(1). 1 xk - 1
In Exercises verify that the function satisfies the differential equation. Function y = 3 cos x + sin x Differential Equation y" + y = 0
Let ƒ(x) = a1 sin x + a2 sin 2x + ...+ an sin nx, where a1, a2,..., an are real numbers and where n is a positive integer. Given that |ƒ(x)| ≤ sin x| for all real x, prove that |a1 + 2a2 + · · · + nan| ≤ 1.
Find the derivative of ƒ(x) = x|x|. Does ƒ"(0) exist?
Let ƒ and g be functions whose first and second derivatives exist on an interval I. Which of the following formulas is (are) true?(a) ƒg" - ƒ"g = (ƒg'- ƒ'g)' (b) ƒg" + ƒ"g = (ƒg)"
Use the Product Rule twice to prove that if ƒ, g, andh are differentiable functions of x, then xp *(x),y(x)8(x)£ + (x)y(x),8(x)£ + (x)y(x)8(x),£ = [(x}y(x)8(x)ƒ]} Р
Prove the difference, product, and quotient properties in Theorem 1.15. THEOREM 1.15 Properties of Infinite Limits Let c and L be real numbers, and let f and g be functions such that lim f(x) = ∞ and lim g(x) = L. 1. Sum or difference: 2. Product: 3. Quotient: lim [f(x) = g(x)] = ∞ lim
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point(b) Use a graphing utility to graph the function and its tangent line at the point(c) Use the derivative feature of a graphing utility to confirm your results f(x) = x³ + 1, (-1,0)
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point(b) Use a graphing utility to graph the function and its tangent line at the point Function y = (x - 2)(x² + 3x) Point (1,-4)
In Exercises determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. y 1 2 X
In Exercises determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.y = x2 + 9
In Exercises the relationship between ƒ and g is given. Explain the relationship between ƒ' and g'.g(x) = ƒ(x) + 6
Use a graphing utility with a square window setting to zoom in on the graph of to approximate ƒ'(1). Use the derivative to find ƒ'(1). zx² - + = (x)ƒ
In Exercises use a graphing utility to graph the function and find the x-values at which ƒ is differentiable. f(x) = x²/5
Use a graphing utility with a square window setting to zoom in on the graph of to approximate ƒ'(4). Use the derivative to findƒ'(4). f(x) = 4√√x + 1
In Exercises find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = |x-1|
In Exercises determine whether the function is differentiable at x = 2. f(x) = [x² + 1, 4x - 3, x ≤ 2 x > 2
In Exercises determine whether the function is differentiable at x = 2. f(x) = fx ²x + 1₂ 1, 2x, x < 2 x ≥ 2
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y = x/π, then dy/dx = 1/π.
In Exercises find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(x): = - 1 X [1, 2]
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If g(x) = 3ƒ(x), then g'(x) = 3ƒ'(x).
In Exercises find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.ƒ(t) = 4t + 5, [1, 2]
In Exercises find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.ƒ(t) = t² - 7, [3, 3.1]
In Exercises find the slope of the tangent line to the graph of the function at the given point. f(x) = 3-5x, (-1,8)
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point(b) Use a graphing utility to graph the function and its tangent line at the point Function f(x) 2 Point (1, 2)
In Exercises(a) Find an equation of the tangent line to the graph of ƒ at the given point(b) Use a graphing utility to graph the function and its tangent line at the point Function y = x³ - 3x Point (2, 2)
In Exercises use a graphing utility to graph the functions ƒ and g in the same viewing window, whereLabel the graphs and describe the relationship between them. g(x) = f(x + 0.01) − f (x) 0.01
In Exercises use the rules of differentiation to find the derivative of the function.y = x7
In Exercises use a graphing utility to graph the functions ƒ and g in the same viewing window, whereLabel the graphs and describe the relationship between them. g(x) = f(x + 0.01) − f (x) 0.01
In Exercises find such that the line is tangent to the graph of the function. Function f(x) = k - x² Line y = -6x + 1
Use a graphing utility to graph each function and its tangent lines at x = -1, x = 0, and x = 1. Based on the results, determine whether the slopes of tangent lines to the graph of a function at different values of x are always distinct.(a) ƒ(x) = x² (b) g(x) = x3
In Exercises find such that the line is tangent to the graph of the function. Function f(x) = kx² Line y = -2x + 3
In Exercises determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.y = x4 -2x² + 3
In Exercises determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.y = x³ + x
In Exercises evaluate ƒ(2) and ƒ(2.1) and use the results to approximate ƒ'(2). εx² = (x)ƒ
In Exercises find such that the line is tangent to the graph of the function. Function k f(x) = X Line y = 3 x + 3
In Exercises evaluate ƒ(2) and ƒ(2.1) and use the results to approximate ƒ'(2). (x - +)x= (x) f
In Exercises use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x)=x²-5, c = 3

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