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mathematics
calculus 10th edition

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

In Exercises determine from the graph whether ƒ has a minimum in the open interval (a, b).(a)(b) y a of b x
In Exercises use symmetry, extrema, and zeros to sketch the graph of ƒ. How do the functions ƒ and g differ? f(t) = cos² t - sin² t g(t) = 1 - 2 sin² t
A plane begins its takeoff at 2:00 P.M. on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.
In Exercises determine from the graph whether ƒ has a minimum in the open interval (a, b).(a)(b) y a of b X
In Exercises the graph of ƒ is shown in the figure. Sketch a graph of the derivative of ƒ. Toprint an enlarged copy of the graph, go to MathGraphs.com. -2 -1 4 2 1 y f 1 2 X
In Exercises determine from the graph whether ƒ has a minimum in the open interval (a, b).(a)(b) y a of b X
When an object is removed from a furnace and placed in an environment with a constant temperature of 90°F, its core temperature is 1500°F. Five hours later, the core temperature is 390°F. Explain why there must exist a time in the interval when the temperature is decreasing at a rate of 222°F
In Exercises the graph of ƒ is shown in the figure. Sketch a graph of the derivative of ƒ. To print an enlarged copy of the graph, go to MathGraphs.com. -2-1 2 1 y If + 1 2 3 X
Two bicyclists begin a race at 8:00 A.M. They both finish the race 2 hours and 15 minutes later. Prove that at some time during the race, the bicyclists are traveling at the same velocity.
The formula for the power output P of a battery is P = VI - RI² where V is the electromotive force in volts, R is the resistancein ohms, and I is the current in amperes. Find the current thatcorresponds to a maximum value of P in a battery for whichV = 12 volts and R = 0.5 ohm. Assume that a
In Exercises , find the absolute extrema of the function on the closed interval. f(x) = 3x, [1, 2]
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = sin x, [0, 2π]
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. 1 ƒ(x)=x²−¹, [-1,1]
In Exercises , find the critical numbers of the function. f(0) = 2 sec 0 + tan 0 0 < 0 < 2TT
In Exercises identify the open intervals on which the function is increasing or decreasing. f(x) = sin²x + sin x, 0 < x < 2π
In Exercises identify the open intervals on which the function is increasing or decreasing. y = x 2 cos x, 0 < x < 2π
In Exercises , find the critical numbers of the function. h(x) = sin² x + cos x 0 < x < 2π
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = x² - 2x - 3 x + 2 [-1,3]
In Exercises identify the open intervals on which the function is increasing or decreasing. h(x) = COS S 2² 0 < x < 2π
In Exercises , find the critical numbers of the function. f(x) 4x x² + 1
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = 3x - 31, [0, 6]
In Exercises identify the open intervals on which the function is increasing or decreasing. f(x) = sin x 1, 0 < x < 2π -
In Exercises , find the critical numbers of the function. g(t) = t√4-t, t < 3
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = x²/31, [-8, 8]
In Exercises identify the open intervals on which the function is increasing or decreasing. y = x + 9 X
In Exercises find the critical numbers of the function.g(x) = x4 - 8x²
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = (x4)(x + 2)², [2, 4]
In Exercises identify the open intervals on which the function is increasing or decreasing. y=x√16 - x²
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = (x - 1)(x-2)(x-3), [1, 3]
In Exercises find the critical numbers of the function.ƒ(x)= x³ - 3x²
In Exercises identify the open intervals on which the function is increasing or decreasing. h(x) = 12x - x³
In Exercises approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown. -2 8 6 4 2 -2. 2 4 6 8
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. f(x) = x² − 8x + 5, [2,6]
In Exercises identify the open intervals on which the function is increasing or decreasing. g(x)=x²- 2x - 8
In Exercises approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown. ㅜㅜ 5 4 3 2 † 1
In Exercises determine whether Rolle's Theorem can be applied to ƒ on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that ƒ'(c) = 0. If Rolle'sTheorem cannot be applied, explain why not. f(x) = -x² + 3x, [0, 3]
In Exercises use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. y = x² 2x - 1 4 3 2 - y -2+1 - + + 23 3 4 X
In Exercises approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown. -1 - 1 У +
In Exercises find the two x-intercepts of the function ƒ and show that ƒ'(x) = 0 at some point between the two x-intercepts. f(x) = −3x√x + 1
In Exercises use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. f(x) = 1 (x + 1)² y -4-3-2-1 1 2 X
In Exercises approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.
In Exercises find the two x-intercepts of the function ƒ and show that ƒ'(x) = 0 at some point between the two x-intercepts. f(x)=x√x + 4
In Exercises use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. f(x) = x² - 2x² 3 نیا 2 1 y 2
In Exercises find the value of the derivative (if it exists) at each indicated extremum. f(x) = 4 = |x| y -2 6 2 -2 (0,4) 2 X
In Exercises use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. || 3 4 -2 4 -4 3x 2 4 X
In Exercises find the two x-intercepts of the function ƒ and show that ƒ'(x) = 0 at some point between the two x-intercepts. f(x) = x² + 6x
In Exercises find the value of the derivative (if it exists) at each indicated extremum. f(x) = (x + 2)2/3 (-2, 0) -4 -3 -2 -1 2 1 -2 y
In Exercises find the two x-intercepts of the function ƒ and show that ƒ'(x) = 0 at somepoint between the two x-intercepts. f(x) = x² - x - 2
In Exercises use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. y = -(x + 1)² y To -3 -1 -1 -2 -3 -4+ X
In Exercises explain why Rolle's Theorem does not apply to the function even though there exist a and b such that ƒ(a) = ƒ(b). f(x) = √(2-x²/³)³, [−1,1]
In Exercises find the value of the derivative (if it exists) at each indicated extremum. f(x) = -3x√√√x + 1 -3 (-3, 2√3)2²- -2 (-1,0) - 1 -2 + 1 X
In Exercises use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. f(x) = x2 – 6x + 8 y 4 3 برا 2 1 - 1 2 4 5
In Exercises find the value of the derivative (if it exists) at each indicated extremum. g(x) = x + 6 5 4 3 2 y (2, 3) 12 4 ++ 3 4 5 6 نرا +x
In Exercises explain why Rolle's Theorem does not apply to the function even though there exist a and b such that ƒ(a) = ƒ(b). ƒ(x) = 1 − |x − 1], [0, 2]
In Exercises use the graph of ƒ to find (a) The largest open interval on which ƒ is increasing(b) The largest open interval on which ƒ is decreasing T y --6-- -4. ----2- 77 -24- H V X
In Exercises find the value of the derivative (if it exists) at each indicated extremum. f(x) = cos 2 -2. TTX 2 (0, 1) 2 (2, -1) 3 X
In Exercises explain why Rolle's Theorem does not apply to the function even though there exist a and b such that ƒ(a) = ƒ(b). X f(x) = cot [77,37] 2'
In Exercises use the graph of ƒ tofind (a) The largest open interval on which ƒ is increasing(b) The largest open interval on which ƒ is decreasing 10 8 6 4 2 2 4 68 10 - X
In Exercises explain why Rolle's Theorem does not apply to the function even though there exist a and b such that ƒ(a) = ƒ(b). f(x) H [−1,1]
In Exercises find the value of the derivative (if it exists) at each indicated extremum. f(x) -2 = x² x² + 4 y 2 1 (0, 0) - 1 -2 12 X
In Exercises the graph of ƒ is shown in the figure. Sketch a graph of the derivative of ƒ. To print an enlarged copy of the graph, go to MathGraphs.com. 2 -4-2 y -4- -6+ 2 f 68 X
At 9:13 A.M., a sports car is traveling 35 miles per hour. Two minutes later, the car is traveling 85 miles per hour. Prove that at some time during this two-minute interval, the car’s acceleration is exactly 1500 miles per hour squared.
A lawn sprinkler is constructed in such a way that dθ/dt is constant, where θ ranges between 45° and 135° (see figure). The distance the water travels horizontally iswhere v is the speed of the water. Find dx/dt and explain why this lawn sprinkler does not water evenly. What part of the lawn
In Exercises the graph of ƒ is shown in the figure. Sketch a graph of the derivative of ƒ. To print an enlarged copy of the graph, go to MathGraphs.com. -6-4 86 6 42 y f ++ 46
The surface area of a cell in a honeycomb isWhere h and s are positive constants and θ is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle θ (π/6 ≤ θ ≤ π/2) that minimizes the surface area S. S = 6hs + 35²√√3- cos 0 sin 0 2
In Exercises the graph of ƒ is shown in the figure. Sketch a graph of the derivative of ƒ. To print an enlarged copy of the graph, go to MathGraphs.com. -4 -2 6 y 4+ -2 f + H 2 4
In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6% (see figure). The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points A and B. The horizontal distances from A
In Exercises the graph of ƒ is shown in the figure. Sketch a graph of the derivative of ƒ. To print an enlarged copy of the graph, go to MathGraphs.com. -4 -2 6 4 2 y -2+ f 2 4
In Exercises assume that ƒ is differentiable for all x. The signs of ƒ' are asfollows.Supply the appropriate inequality sign for the indicated value of c. f'(x) > 0 on (-∞, - 4) f'(x) < 0 on (-4, 6) f'(x) > 0 on (6, ∞)
In Exercises sketch the graph of an arbitrary function ƒ that satisfies the given condition butdoes not satisfy the conditions of the Mean Value Theorem onthe interval [-5, 5].ƒ is continuous on [-5, 5].
In Exercises assume that ƒ is differentiable for all x. The signs of ƒ' are as follows.Supply the appropriate inequality sign for the indicated value of c. f'(x) > 0 on (-∞, -4) f'(x) < 0 on (-4, 6) f'(x) > 0 on (6, ∞)
In Exercises sketch the graph of an arbitrary function ƒ that satisfies the given condition but does not satisfy the conditions of the Mean Value Theorem on the interval [-5, 5].ƒ is not continuous on [-5, 5].
In Exercises assume that ƒ is differentiable for all x. The signs of ƒ' are as follows.Supply the appropriate inequality sign for the indicated value of c. f'(x) > 0 on (-∞, -4) f'(x) < 0 on (-4, 6) f'(x) > 0 on (6, ∞)
In Exercises use the Intermediate Value Theorem and Rolle’s Theorem to prove that the equation has exactly one real solution. 0 = 1 + x + εX + s.X
In Exercises assume that ƒ is differentiable for all x. The signs of ƒ' are as follows.Supply the appropriate inequality sign for the indicated value of c. f'(x) > 0 on (-∞, -4) f'(x) < 0 on (-4, 6) f'(x) > 0 on (6, ∞)
In Exercises use the Intermediate Value Theorem and Rolle’s Theorem to prove that the equation has exactly one real solution. 2x³ + 7x1 = 0
In Exercises assume that ƒ is differentiable for all x. The signs of ƒ' are as follows.Supply the appropriate inequality sign for the indicated value of c. f'(x) > 0 on (-∞, -4) f'(x) < 0 on (-4, 6) f'(x) > 0 on (6, ∞)
In Exercises use the Intermediate Value Theorem and Rolle’s Theorem to prove that the equation has exactly one real solution. 3x + 1 sin x = 0
Let the function ƒ be differentiable on an interval I containing c. If ƒ has a maximum value at x = c, show that -ƒ has a minimum value at x = c.
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.The Mean Value Theorem can be applied toon the interval [-1, 1]. f(x) = 1 X
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 the graph of a function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal.
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 the graph of a polynomial function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal.
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 ƒ'(x) = 0 for all x in the domain of ƒ, then ƒ is a constantfunction.
In Exercises find a function ƒ that has the derivative ƒ'(x) and whose graph passes through the given point. Explain your reasoning. f'(x) = 0, (2,5)
In Exercises assume that ƒ is differentiable for all x. The signs of ƒ' are as follows.Supply the appropriate inequality sign for the indicated value of c. f'(x) > 0 on (-∞, -4) f'(x) < 0 on (-4, 6) f'(x) > 0 on (6, ∞)
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.The relative maxima of the function ƒ are ƒ(1) = 4 and ƒ(3) = 10. Therefore, ƒ has at least one minimum for some xin the interval (1, 3).
In Exercises use the Intermediate Value Theorem and Rolle’s Theorem to prove that the equation has exactly one real solution. 2x2 cos x = 0
Sketch the graph of the arbitrary function ƒ such that > 0, f'(x) undefined, < 0, x < 4 x = 4. x > 4
Use the graph of ƒ' to (a) Identify the critical numbers of ƒ(b) Identify the open interval(s) on which ƒ isincreasing or decreasing(c) Determinewhether ƒ has a relative maximum, a relativeminimum, or neither at each critical number(i)(ii)(iii)(iv) F -2 2 -2 -4 y 2 || > x 4
In Exercises find a function ƒ that has the derivative ƒ'(x) and whose graph passes through the given point. Explain your reasoning. f'(x) = 2x, (1, 0)
In Exercises find a function ƒ that has the derivative ƒ'(x) and whose graph passes through the given point. Explain your reasoning. f'(x) = 4, (0, 1)
Determine all real numbers a > 0 for which there exists a nonnegative continuous function ƒ(x) defined on [0, a] with the property that the region R = {(x, y); 0 ≤ x ≤ a, 0 ≤ y ≤ ƒ(x)} has perimeter k units and area k squareunits for some real number k.
In Exercises the function ƒ is differentiable on the indicated interval. The table shows ƒ'(x) for selected values of x. (a) Sketch the graph of ƒ(b) Approx-mate the critical numbers(c) Identify the relative extremaƒ is differentiable on [0, π]. 0 f'(x) 3.14 X X TT/6 - 0.23 2/3
In Exercises find a function ƒ that has the derivative ƒ'(x) and whose graph passes through the given point. Explain your reasoning. f'(x) = 6x 1, (2,7) -
In Exercises the function ƒ isdifferentiable on the indicated interval. The table shows ƒ'(x)for selected values of x. (a) Sketch the graph of ƒ(b) Approx-mate the critical numbers(c) Identify the relative extremaƒ is differentiable on [−1, 1]. f'(x) -1 - 10 0.25 -0.75 -0.50
A differentiable function ƒ has one critical number at x = 5. Identify the relative extrema of ƒ at the critical number when ƒ'(4) = -2.5 and ƒ'(6) = 3.
A differentiable function ƒ has one critical number at x = 2. Identify the relative extrema of ƒ at the critical number when ƒ'(1) = 2 and ƒ'(3) = 6.
The concentration C of a chemical in the bloodstream hours after injection into muscle tissue is(a) Complete the table and use it to approximate the time when the concentration is greatest.(b) Use a graphing utility to graph the concentration function and use the graph to approximate the time when
Coughing forces the trachea (windpipe) to contract, which affects the velocity of the air passing through the trachea. The velocity of the air during coughing iswhere is a constant, is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air
The resistance of a certain type of resistor iswhere R is measured in ohms and the temperature T is measured in degrees Celsius.(a) Use a computer algebra system to find dR/dT and the critical number of the function. Determine the minimum resistance for this type of resistor.(b) Use a graphing

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