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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

In Exercises use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. f(x) = 2 sin 2x X
In Exercises use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. g(x) = 2x √3x² + 1
In Exercises use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. g(x) = sin( X x-2, x > 3
In Exercises (a) Use a graphing utility to graph ƒ and g in the same viewing window(b) Verify algebraically that ƒ and g represent the samefunction (c) Zoom out sufficiently far so that
Use the definition of infinite limits at infinity to prove that lim x3 = = ∞. ∞o. X=
In Exercises use the definition of limits at infinity to prove the limit. 1 lim x-xx-2 = 0 =
In Exercises use the definition of limits at infinity to prove the limit. 1 lim x118 13 = 0
In Exercises use the definition of limits at infinity to prove the limit. lim 2 X/ = 0
In Exercises use the definition of limits at infinity to prove the limit. 1 lim = 0 x-00x²
The average typing speeds S (in words per minute) of a typing student after t weeks of lessons are shown in the table.A model for the data is(a) Use a graphing utility to plot the data and graph the
First Law of Motion and Einstein's Special Theory of Relativity differ concerning a particle's behavior as its velocity approaches the speed of light c. In the graph, functions N and E represent the
A business has a cost of C = 0.5x + 500 for producing x units. The average cost per unit isFind the limit of C̅ as approaches infinity. IN C || C X
The efficiency of an internal combustion engine is Efficiencywhere V1/V2 is the ratio of the uncompressed gas to the compressed gas and is a positive constant dependent on the engine design.
In Exercises (a) Use a graphing utility to graph ƒ and g in the same viewing window(b) Verify algebraically that ƒ and g represent the same function (c) Zoom out sufficiently far so that
A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. The results for the first 2 minutes are
(a) Find an equation of the normal line to the ellipseat the point (4, 2). (b) Use a graphing utility to graph the ellipse and the normal line. (c) At what other point does the normal line
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 real numbers x, then ƒ decreases
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
In Exercises find all relative extrema. Use the Second Derivative Test where applicable. f(x) = x + 4 X
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
In Exercises use a graphing utility to (a) Graph the function ƒ on the given interval (b) Find and graph the secant line through points on the graph of ƒ at the endpoints of the given
Show that the point of inflection oflies midway between the relative extrema of ƒ. f(x) = x(x - 6)²
In Exercises let ƒ and g represent differentiable functions such that ƒ" # 0 and g" # 0.Prove that if ƒ and g are positive, increasing, and concave upward on the interval (a, b), then ƒg is also
In Exercises use a graphing utility to graph the function. Then graph the linear and quadratic approximationsin the same viewing window. Compare the values of ƒ, P1, and P2 and their first
In Exercises let ƒ and g represent differentiable functions such that ƒ" # 0 and g" # 0.Show that if ƒ and g are concave upward on the interval (a, b), then ƒ + g is also concave upward on (a, b).
In Exercises use a graphing utility to graph the function. Then graph the linear and quadratic approximationsin the same viewing window. Compare the values of ƒ, P1, and P2 and their first
In Exercises use a graphing utility to graph the function. Then graph the linear and quadratic approximationsin the same viewing window. Compare the values of ƒ, P1, and P2 and their first
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"(2) = 0, then the graph of ƒ must have a point of
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'(c) > 0, then ƒ is concave upward at x = c.
A small aircraft starts its descent from an altitude of 1 mile, 4 miles west of the runway (see figure).(a) Find the cubic ƒ(x) = ax³ + bx² + cx + d on the interval [-4, 0] that describes a smooth
In Exercises use a graphing utility to graph the function. Then graph the linear and quadratic approximationsin the same viewing window. Compare the values of ƒ, P1, andP2 and their first
A manufacturer has determined that the total cost C of operating a factory iswhere X is the number of units produced. At what level of production will the average cost per unit be minimized? (The
A section of highway connecting two hillsides with grades of 6% and 4% is to be built between two points that are separated by a horizontal distance of 2000 feet (see figure). At the point where the
Consider the function(a) Graph the function and identify the inflection point.(b) Does f"(x) exist at the inflection point? Explain. f(x) = 3√x. X.
Consider the function(a) Use a graphing utility to graph ƒ for n = 1, 2, 3, and 4. Use the graphs to make a conjecture about the relationship between n and any inflection points of the graph of
In Exercises find a, b, c, and d such that the cubicsatisfies the given conditions.Relative maximum: (2, 4)Relative minimum: (4, 2)Inflection point: (3, 3) f(x) = ax³ + bx² + cx + d
In Exercises find a, b, c, and d such that the cubicsatisfies the given conditions.Relative maximum: (3, 3)Relative minimum: (5, 1)Inflection point: (4, 2) f(x) = ax³ + bx² + cx + d
In Exercises, sketch the graph of a function having the given characteristics. f(0) = f(2)= 0 f'(x) < 0 for x < 1 f'(1) = 0 f'(x) > 0 for x > 1 f"(x) > 0
Water is running into the vase shown in the figure at a constant rate.(a) Graph the depth of water in the vase as a function of time.(b) Does the function have any extrema? Explain.(c) Interpret the
The figure shows the graph of ƒ". Sketch a graph of ƒ. To print an enlargedcopy of the graph, go to MathGraphs.com. 5432 1 - 1 y f" 1 2 3 4 5
In Exercises, sketch the graph of a function having the given characteristics. f(2)=f(4) = 0 f'(x) < 0 forx < 3 f'(3) does not exist. f'(x) > 0 forx > 3 f"(x) < 0, x = 3
In Exercises, sketch the graph of a function having the given characteristics. f(2)=f(4) = 0 f'(x) > 0 forx 3 f"(x) > 0, x = 3
In Exercises (a) Use a computer algebra system to differentiate the function(b) Sketch the graphs of ƒ and ƒ' on the same set of coordinate axes over the given interval(c) Find the critical
In Exercises, sketch the graph of a function having the given characteristics. f(0) = f(2)= 0 f'(x) > 0 for x < 1 f'(1) = 0 f'(x) < 0 for x > 1 f"(x) < 0
Determine whether each labeled point is an absolute maximum or minimum, a relative maximum or minimum, or none of these. y B E C AN F D A G
In Exercises the graph of ƒ is shown. Graph ƒ, ƒ', and ƒ" on the same set of coordinate axes. To print an enlarged copy of the graph, go to MathGraphs.com. f -2 4 -2 -4 y 1 2 X
In Exercises (a) Use a computer algebra system to differentiate the function(b) Sketch the graphs of ƒ and ƒ' on the same set of coordinate axes over the given interval(c) Find the critical
In Exercises (a) Use a computer algebra system to differentiate the function(b) Sketch the graphs of ƒ and ƒ' on the same set of coordinate axes over the given interval(c) Find the critical
In Exercises the graph of ƒ is shown. Graph ƒ, ƒ', and ƒ" on the same set of coordinate axes. Toprint an enlarged copy of the graph, go to MathGraphs.com. T -1 3 12 y -14 1/2 + 3 X
In Exercises use a computer algebra system to find the maximum value of |ƒ(4) (x)| on the closed interval. f(x) = 1 x² + 1' [-1,1]
Write a short paragraph explaining why a continuous function on an open interval may not have a maximum or minimum. Illustrate your explanation with a sketch of the graph of such a function.
In Exercises (a) Use a computer algebra system todifferentiate the function(b) Sketch the graphs of ƒ and ƒ' onthe same set of coordinate axes over the given interval(c) Find the critical
In Exercises use a computer algebra system to find the maximum value of |ƒ(4) (x)| on the closed interval. f(x) = (x + 1)2/3, [0, 2]
In Exercises use a graphing utility to (a) Graph the function ƒ on the given interval (b) Find and graph the secant line through points on the graph of ƒ at the endpoints of the given
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
In Exercises use a graphing utility to (a) Graph the function ƒ on the given interval (b) Find and graph the secant line through points on the graph of ƒ at the endpoints of the given
If S represents weekly sales of a product. What can be said of S' and S" for each of the following statements? (a) The rate of change of sales is increasing. (b) Sales are increasing at a
In Exercises use a computer algebra system to find the maximum value of ƒ"(x) on the closed interval. f(x) = 1 x² + 1² 11/12, 3
Sketch the graph of a function ƒ that does not have a point of inflection at (c, ƒ(c)) even though ƒ"(c) = 0.
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
In Exercises use a computer algebra system to find the maximum value of |ƒ"(x)| on the closed interval. f(x) = √√√1 + x³, [0,2]
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
Consider a function ƒ such that ƒ' is decreasing. Sketch graphs of ƒ for (a) ƒ' < 0 (b) ƒ' > 0.
In Exercises use a graphing utility to (a) Graph the function ƒ on the giveninterval (b) Find and graph the secant line through points on the graph of ƒ at the endpoints of the given
In Exercises(a) Use a computer algebra system to graph the function and approximate any absolute extrema on the given interval.(b) Use the utility to find any critical numbers, and use them to find
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
In Exercises use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and
Graph Consider a function ƒ such that ƒ' is increasing. Sketch graphs of ƒ for (a) ƒ' < 0 (b) ƒ' > 0.
In Exercises determine 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)
In Exercises use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and
In Exercises determine 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)
In Exercises(a) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval.(b) Use the utility to find any critical numbers, and use them to find
In Exercisesπ consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
In Exercises use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. f(x) = √x + cos, [0, 2π]
In Exercises use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
In Exercises use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. f(x) = x² - 2x³ + x + 1, [-1,3]
In Exercises determine 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)
In Exercises determine 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)
In Exercises use a computer algebra system to analyze the function over the given interval. (a) Findthe first and second derivatives of the function. (b) Find anyrelative extrema and points
In Exercises find all relative extrema. Use the Second Derivative Test where applicable. f(x) = 2 sin x + cos 2x, [0, 2π]
In Exercises determine 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)
In Exercises use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. f (x) 2 2-x² [0, 2)
In Exercises use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. f(x) = 3 x - l' (1,4]
In Exercises determine 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)
In Exercises π consider the function on the interval (0, 2π). For each function(a) Find the open interval(s) on which the function is increasing or decreasing (b) Apply the First Derivative
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
In Exercises find the absolute extrema of the function (if any exist) on each interval.(a) [-2, 2] (b) [-2, 0)(c) (-2, 2)(d) [1, 2) f(x)=√√√4x²
In Exercises find all relative extrema. Use the Second Derivative Test where applicable. f(x) = cos xx, [0, 4π]
In Exercises find the absolute extrema of the function (if any exist) on each interval.(a) [-1, 2](b) (1, 3](c) (0, 2)(d) [1, 4) f(x) = x² - 2x
In Exercises determine 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)
In Exercises determine 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)
In Exercises find all relative extrema. Use the Second Derivative Test where applicable. f(x) = X x - 1
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
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
In Exercises find the absolute extrema of the function (if any exist) on each interval.(a) [1, 4] (b) [1, 4)(c) (1, 4] (d) (1, 4) f(x) = 5 - x
In Exercises determine 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)
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
In Exercises find the absolute extrema of the function (if any exist) on each interval.(a) [0, 2](b) [0, 2)(c) (0, 2](d) (0, 2) f(x) = 2x - 3
In Exercises find all relative extrema. Use the Second Derivative Test where applicable. f(x)=√√√x² + 1

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