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mathematics
calculus 10th edition

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

In Exercises complete the table to find the indefinite integral. Original Integral √(3x)2 dx Rewrite Integrate Simplify
In Exercises use sigma notation to write the sum. () - () +----[(%) - () 3 2 " 3 2n +. n l
In Exercises write the limit as a definite integral on the interval [a, b], where ci is any point in the ith subinterval. Limit Interval lim Σ 6c;(4 – c;)2 Δx [0, 4] ||Δ||-0 1
In Exercises write the limit as a definite integral on the interval [a, b], where ci is any point in the ith subinterval. Limit n 2 lim Σvc? + 4 Δx ||Δ||−0 Interval [0, 3]
In Exercises find the indefinite integral and check the result by differentiation. fax + (x + 7) dx
In Exercises use sigma notation to write the sum. 3 3 [ 2(1 + * (2) n n 3n +. --- 3)10) n n +21+
In Exercises write the limit as a definite integral on the interval [a, b], where ci is any point in the ith subinterval. Limit lim (²) 4x 3 2 ||A||→0 Interval [1, 3]
In Exercises find the indefinite integral and check the result by differentiation. [(13 - x) dx
In Exercises, set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 6 - 3x 6 543 4 -2-1 2 1 y 11 1 2 3 4 5 X
In Exercises, set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 5 y 5 4 3 2 1 1 2 3 4 5 X
In Exercises use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result.Data from in Theorem 4.2 12 Στ i=1
In Exercises use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result.Data from in Theorem 4.2 30 Σ-18 i=1
In Exercises find the indefinite integral and check the result by differentiation. [6x3/2 (x3/2 + 2x + 1) dx
In Exercises find the indefinite integral and check the result by differentiation. [(8) (8x³9x² + 4) dx
In Exercises, set up a definite integral that yields the area of the region. (Do not evaluate the integral.) X 4 2 2 -9 y 8 00 てーヤー 4 || -v = (x) f
In Exercises use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result.Data from in Theorem 4.2 16 i=1 (5i - 4)
In Exercises use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result.Data from in Theorem 4.2 24 i=1 4i
In Exercises, set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 25 - x² 15 10 5 -6-4-2 + 246 X
In Exercises, set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = x2 -1 4 نا 3 2 1 1 2 3 X
In Exercises find the indefinite integral and check the result by differentiation. 1 √(√x + 2√ =) dx
In Exercises use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result.Data from in Theorem 4.2 20 Σ( − 1)2 i=1
In Exercises find the indefinite integral and check the result by differentiation. S 3/x² dx
In Exercises, set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) 4 x² + 2 1
Graph the fourth degree polynomialfor various values of the constant a.(a) Determine the values of a for which p has exactly one relative minimum.(b) Determine the values of a for which p has exactly one relative maximum.(c) Determine the values of a for which p has exactly two relative minima.(d)
LetDetermine all values of the constant c such that ƒ has a relativeminimum, but no relative maximum. C f(x) = =+ x². X
Prove the following Extended Mean Value Theorem. If ƒ and ƒ' are continuous on the closed interval [a, b], and if ƒ" exists in the open interval(a, b), then there exists a number c in (a, b) such that ƒ(b) = f(a) + f'(a)(b − a) + − ƒ”(c)(b − a)². - -
The figures show a rectangle, a circle, and a semicircle inscribed in a triangle bounded by the coordinate axes and the first-quadrant portion of the line with intercepts (3, 0) and (0, 4). Find the dimensions of each inscribed figure such that its area is maximum. State whether calculus was
The amount of illumination of a surface is proportional to the intensity of the light source, inversely proportional to the square of the distance from the light source, and proportional to sin θ, where θ is the angle at which the light strikes the surface. A rectangular room measures 10 feet by
Consider a room in the shape of a cube, 4 meters on each side. A bug at point P wants to walk to point Q at the opposite corner, as shown in the figure. Use calculus to determine the shortest path. Explain how you can solve this problem without calculus. P 4 m 4 m 4 m Q
Determine the values a, b, and c such that the function ƒ satisfies the hypotheses of the MeanValue Theorem on the interval [0, 3]. f(x) = ax + b, x² + 4x + c, x = 0 0 < x≤ 1 1 < x ≤ 3
The line joining P and Q crosses the two parallel lines, as shown in the figure. The point R is d units from P. How far from Q should the point S be positioned so that the sum of the areas of the two shaded triangles is a minimum? So that the sum is a maximum? P S d R
Determine the values a, b, c, and d such that the function ƒ satisfies the hypotheses of the MeanValue Theorem on the interval [-1, 2]. f(x) = = a, 2, bx² + c₂ dx + 4, x = -1 -1 < x≤ 0 0 < x ≤ 1 1 < x≤ 2
Let f ƒ and g be functions that are continuous on [a, b] and differentiable on (a, b). Prove that if ƒ(a) = g(a) and g'(x) > ƒ'(x) for all x in (a, b), then g(b) > ƒ(b).
Let ƒ be differentiable on the closed interval [a, b] such that ƒ'(a) = y₁ and ƒ'(b) = y₂. If d lies between y1, and y2, then there exists c in (a, b) such that ƒ'(c) = d.
Show that the cubic polynomial p(x) = ax³ + bx² + cx + d has exactly one point of inflection (x0, y0), whereUse this formula to find the point of inflection of p(x) = x³ - 3x² + 2. ХО = -b За and Yo = 263 27a² bc За + d.
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. h(t) 1 -14 - 8t
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. g(x) = x³8x 4
In Exercises, find all relative extrema. Use the Second Derivative Test where applicable. f(x) = 2x³ + 11x² - 8x - 12
In Exercises sketch the graph of a function ƒ having the given characteristics. ƒ(0) = 4, ƒ(6) = 0 f'(x) < 0 for x < 2 or x > 4 f'(2) does not exist. f'(4) = 0 f'(x) > 0 for 2 < x < 4 f"(x) < 0 for x # 2
In Exercises, find the absolute extrema of the function on the closed interval. f(x)=√x-2, [0,4]
In Exercises, find the absolute extrema of the function on the closed interval. f(x) = x² + 5x, [-4,0]
In Exercises, find the absolute extrema of the function on the closed interval. h(x) = 3√√√x-x, [0,9]
In Exercises, find the absolute extrema of the function on the closed interval. f(x) = x³ + 6x², [-6,1]
In Exercises, find the absolute extrema of the function on the closed interval. g(x) = 2x + 5 cos x, [0, 2π]
In Exercises, find the absolute extrema of the function on the closed interval. f(x) = sin 2x, [0, 2π]
In Exercises, find the absolute extrema of the function on the closed interval. f(x) = 4x x² + 9' [-4,4]
In Exercises, find the absolute extrema of the function on the closed interval. f(x) = X x² + 1 2 [0, 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) = x² 1 - x²² [-2,2]
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. IfRolle's Theorem cannot be applied, explain why not. f(x) = 2x²7, [0, 4]
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-2)(x + 3)², [-3,2]
In Exercises detrmine whether the Mean Value Theorem can be applied to ƒ on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = ƒ(b) – ƒ(a) b - a
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 2x, [-π, π]
In Exercises detrmine whether the Mean Value Theorem can be applied to ƒ onthe closed interval [a, b]. If the Mean Value Theorem can beapplied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = - ƒ(b) – ƒ(a) b - a
In Exercises detrmine whether the Mean Value Theorem can be applied to ƒ on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = ƒ(b) – ƒ(a) b - a
In Exercises detrmine whether the Mean Value Theorem can be applied to ƒ on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = ƒ(b) – ƒ(a) b - a
In Exercises detrmine whether the Mean Value Theorem can be applied to ƒ on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = ƒ(b) – ƒ(a) b - a
In Exercises detrmine whether the Mean Value Theorem can be applied to ƒ on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = ƒ(b) – ƒ(a) b - a
In Exercises, identify the open intervals on which the function is increasing or decreasing. f(x) = x² + 3x - 12
Can the Mean Value Theorem be applied to the functionon the interval [-2, 1]? Explain. f(x) 뚜뚜 ||
In Exercises, identify the open intervals on which the function is increasing or decreasing. h(x) = (x + 2)1/3 + 8
In Exercises, identify the open intervals on which the function is increasing or decreasing. g(x) = (x + 1)³
In Exercises, identify the open intervals on which the function is increasing or decreasing. f(x) = (x - 1)²(x − 3) -
(a) For the function ƒ(x) = Ax² + Bx + C, determine the value of c guaranteed by the Mean Value Theorem on the interval [x₁, x₂].(b) Demonstrate the result of part (a) for ƒ(x) = 2x² - 3x + 1on the interval [0, 4].
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relativeextrema (d) Use a graphing utility to confirm your results. f(x) = x² - 6x + 5
In Exercises, identify the open intervals on which the function is increasing or decreasing. 0 < x (EX)x/^ = (x)u
In Exercises, identify the open intervals on which the function is increasing or decreasing. f(x) = sin x + cos x, [0, 2π]
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. f(x) = x + 4 x²
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. f(x) = 4x³ - 5x
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. x² - 3x - 4 x - 2
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. f(x) = cos x - sin x, (0, 2TT)
In Exercises(a) Find the critical numbers of ƒ (if any) (b) Find the open interval(s) on which the function is increasing or decreasing(c) Apply the First Derivative Test to identify all relative extrema (d) Use a graphing utility to confirm your results. g(x) π.Χ. (7² - 1), 2 3 =
In Exercises, find the points of inflection and discuss the concavity of the graph of the function. f(x) = x³9x²
In Exercises, find the points of inflection and discuss the concavity of the graph of the function. f(x) = tan (0, 2)
In Exercises, find the points of inflection and discuss the concavity of the graph of the function. f(x) = 6x4x²
In Exercises, find the points of inflection and discuss the concavity of the graph of the function. g(x)=x√√√x + 5
In Exercises, find the points of inflection and discuss the concavity of the graph of the function. f(x) = 3x - 5x3 5x³
In Exercises, find the points of inflection and discuss the concavity of the graph of the function. f(x) = x + cos x, [0, 2π]
In Exercises, find all relative extrema. Use the Second Derivative Test where applicable. f(x) = (x + 9)²
In Exercises, find all relative extrema. Use the Second Derivative Test where applicable. g(x) = 2x²(1-x²)
In Exercises, find all relative extrema. Use the Second Derivative Test where applicable. h(t) = t - 4√t + 1
In Exercises, find all relative extrema. Use the Second Derivative Test where applicable. f(x) = 2x + 18 X
In Exercises, find all relative extrema. Use the Second Derivative Test where applicable. h(x) = x - 2 cos x, [0, 4π]
In Exercises sketch the graph of a function ƒ having the given characteristics. f(0) = f(6) = 0 f'(3) = f'(5) = 0 f'(x) > 0 for x < 3 f'(x) > 0 for 3 < x < 5 f'(x) < 0 for x > 5 f"(x) < 0 for x < 3 or x > 4 f"(x) > 0 for 3 < x < 4
The cost of inventory C depends on the ordering and storage costs according to the inventory modelDetermine the order size that will minimize the cost, assuming that sales occur at a constant rate, Q is the number of units sold per year, r is the cost of storing one unit for one year, s is the cost
In Exercises, find the limit. lim 8 + x-∞ X
In Exercises, find the limit. lim x-∞0 3x² x + 5
In Exercises, find the limit. lim x-∞0 1 - 4x x + 1
In Exercises, find the limit. lim x →∞ 2x² 3x² + 5
In Exercises, find the limit. 4x³ lim xx0x² + 3
In Exercises, find the limit. lim ∞07-x x³ x² + 2
A newspaper headline states that “The rate of growth of the national deficit is decreasing.” What does this mean? What does it imply about the graph of the deficit as a function of time?
In Exercises, find the limit. lim 8118 x + zx₁ - 2x
In Exercises, find the limit. lim x →∞ 5 cos x X
In Exercises, find the limit. 6x lim x-∞0 x + cos x
In Exercises, use a graphing utility to graph the function and identify any horizontal asymptotes. f(x) = 3x √x² + 2
In Exercises, find the limit. lim x→→∞ 2 sin x 8118
In Exercises, use a graphing utility to graph the function and identify any horizontal asymptotes. g(x) = 5x² x² + 2
In Exercises, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. f(x)=x√16 - x²
In Exercises, use a graphing utility to graph the function and identify any horizontal asymptotes. h(x) = 2x + 3 x - 4
In Exercises, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. f(x) = 4x - x²

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