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mathematics
calculus 10th edition
Questions and Answers of
Calculus 10th Edition
In Exercises show that ƒ and g are inverse functions (a) Analytically (b) Graphically. x 91 = (x)8 0 = x x 91 = (x)ƒ
Prove that the function is constant on the interval (0, ∞). F(x) *2x [²1/ x = - dt
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. 1 1/ 4x = dx = [ 1n/x1] ²₁ = - 1 In 2 In 1 In 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. f/=/dx = 1 dx = In|cx|, c = 0
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. fIn x dx = (1/x) + C
In Exercises show that ƒ and g are inverse functions (a) Analytically (b) Graphically. f(x) = x³, g(x) 3, =
In Exercises show that ƒ and g are inverse functions (a) Analytically (b) Graphically. f(x) = 3 - 4x, g(x) = 3 x 4
For 0 < x < y, show that 1 y V In y - In x y - x 1
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. (In x)¹/2 = In x
LetShow that y is its own inverse function. What can you conclude about the graph of ƒ? Explain. y || x x - 2 1'
In Exercises find the average value of the function over the given interval. f(x) = = 2 ln x X [1, e]
In Exercises use logarithmic differentiation to find dy/dx. y (x + 1)(x - 2) (x - 1)(x + 2)' x > 2
In Exercises, find the indefinite integral. 1110 dx - X
Archimedes showed that the area of a parabolic arch is equal to 2/3 the product of the base and the height (see figure).(a) Graph the parabolic arch bounded by y = 9 - x² and the x-axis. Use an
Let(a) Find L(1).(b) Find L'(x) and L'(1).(c) Use a graphing utility to approximate the value of x (to three decimal places) for which L(x) = 1.(d) Prove that L(x1x2) = L(x1) + L(x2) for all positive
In Exercises(a) Write the area under the graph of the given function defined on the given interval as a limit. (b) Evaluate the sum in part (a).(c) Evaluate the limit using the result of part
In Exercises(a) Write the area under the graph of the given function defined on the given interval as a limit. (b) Evaluate the sum in part (a).(c) Evaluate the limit using the result of part
The Two-Point Gaussian Quadrature Approximation for ƒ is(a) Use this formula to approximateFind the error of the approximation.(b) Use this formula to approximate(c) Prove that the Two-Point
Galileo Galilei (1564–1642) stated the following proposition concerning falling objects:The time in which any space is traversed by a uniformly accelerating body is equal to the time in which
Use an appropriate Riemann sum to evaluate the limitProve f(x)f'(x) dx = ¹/([ƒ(b)]² − [ƒ(a)}²). -
Use an appropriate Riemann sum to evaluate the limitProve ["* (D(x - 1) de = [" ("* (v) adv ) dt. f(t)(x-t) dt - 0 0
Suppose that ƒ is integrable on [a, b] and 0 < m ≤ ƒ(x) ≤ M for all x in the interval [a, b]. Prove thatUse this result to estimate m(a - b) ≤ b [ f(x) dx ≤ M(b − a).
Use an appropriate Riemann sum to evaluate the limit z/c U . . u^ + · · · + £/ +7/ + I/ olu ալ
Use an appropriate Riemann sum to evaluate the limit lim 810 15 + 25 + 35 + + n² n6 .
Suppose that ƒ is integrable on [a, b] and 0 < m ≤ ƒ(x) ≤ M for all x in the interval [a, b]. Prove thatUse this result to estimate m(a - b) ≤ b [ f(x) dx ≤ M(b − a).
A car travels in a straight line for 1 hour. Its velocity in miles per hour at six-minute intervals is shown in the table.(a) Produce a reasonable graph of the velocity function v by graphing these
Prove that if ƒ is a continuous function on a closed interval [a, b], then b |[²*1(x) dx = [ * 15(6x) dx.
A car travels in a straight line for 1 hour. Its velocity in miles per hour at six-minute intervals is shown in the table.(a) Produce a reasonable graph of the velocity function v by graphing these
Verify thatby showing the following.(a)(b)(c) n Στ i=1 – n(n + 1)(2n + 1) 6
In Exercises, find the indefinite integral. - dx X
Determine the limits of integration where a ≤ b such thathas minimal value. So (x² - 16) dx
In Exercises, find the indefinite integral. 1 +1 dx
In Exercises use a graphing utility to evaluate the logarithm by (a) Using the naturallogarithm key (b) Using the integration capabilities toevaluate the integral ∫1x(1/t) dt.In 45
In Exercises match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b)(c)(d) 2 -1 -2 -3 y ++ 2 3 4 5 -X
In Exercises, find the indefinite integral. 1 Jx-5 dx
In Exercises use a graphing utility to evaluate the logarithm by (a) Using the natural logarithm key (b) Using the integration capabilities to evaluate the integral ∫1x(1/t) dt.In 8.3
In Exercises, find the indefinite integral. 9 5 - 4x dx
In Exercises, find the indefinite integral. 1 2x + 5 dx
In Exercises match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b)(c)(d) 2 -1 -2 -3 y ++ 2 3 4 5 -X
In Exercises use a graphing utility to evaluate the logarithm by (a) Using the natural logarithm key (b) Using the integration capabilities to evaluate the integral ∫1x(1/t) dt.In 0.8
In Exercises, find the indefinite integral. X x² - 3 dx
In Exercises use a graphing utility to evaluate the logarithm by (a) Using the natural logarithm key (b) Using the integration capabilities to evaluate the integral ∫1x(1/t) dt.In 0.6
In Exercises match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b)(c)(d) 2 -1 -2 -3 y ++ 2 3 4 5 -X
In Exercises, find the indefinite integral. x² 5x³ dx
In Exercises match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b)(c)(d) 2 -1 -2 -3 y ++ 2 3 4 5 -X
In Exercises, find the indefinite integral. 4x³+3 x4 + 3x dx
In Exercises, sketch the graph of the function and state its domain. f(x) = 3 ln x
In Exercises, find the indefinite integral. x² - 2x x³ - 3x² dx
In Exercises, sketch the graph of the function and state its domain. f(x) = -2 ln x
In Exercises, sketch the graph of the function and state its domain. f(x) = In 2x
In Exercises, find the indefinite integral. x² - 4 X dx
In Exercises, find the indefinite integral. - 8x x² dx
In Exercises, find the indefinite integral. x² + 2x + 3 x³ + 3x² + 9x 3 dx
In Exercises, sketch the graph of the function and state its domain. f(x) = ln (x − 3) -
In Exercises, sketch the graph of the function and state its domain. f(x) = In[x]
In Exercises, find the indefinite integral. x² + 4x x³ + 6x² + 5 dx
In Exercises, sketch the graph of the function and state its domain. f(x) = ln x - 4
In Exercises, find the indefinite integral. x² - 3x + 2 x + 1 dx
In Exercises, sketch the graph of the function and state its domain. h(x) = ln(x + 2)
In Exercises, find the indefinite integral. [2x² + 7x - 3 x 2 16 dx
In Exercises, sketch the graph of the function and state its domain. f(x) = ln(x - 2) + 1
In Exercises, find the indefinite integral. fr²³. - 3x² + 5 x - 3 dx
In Exercises use the properties of logarithms to approximate the indicated logarithms, given that In 2 ≈ 0.6931 and In 3 ≈ 1.0986.(a) In 6(b) In 2/3(c) In 81(d) In √3
In Exercises, find the indefinite integral. x3 - 6x - 20 -3 x + 5 dx
In Exercises, find the indefinite integral. x4 + x - 4 x² x² + 2 - dx
In Exercises, use the properties of logarithms to expand the logarithmic expression. In X 4
In Exercises, find the indefinite integral. x³ 4x² - 4x + 20 x² - 5 dx
In Exercises, use the properties of logarithms to expand the logarithmic expression. In√x5
In Exercises use the properties of logarithms to approximate the indicated logarithms, given that In 2 ≈ 0.6931 and In 3 ≈ 1.0986.(a) In 0.25(b) In 24(c) In 3√12(d) In 1/72
In Exercises, find the indefinite integral. [(In x)² dx
In Exercises, find the indefinite integral. 1 √x(1-3√√√x) dx
In Exercises, use the properties of logarithms to expand the logarithmic expression. ху In 2
In Exercises, find the indefinite integral. 1 x ln x³ dx
In Exercises, use the properties of logarithms to expand the logarithmic expression. In(xyz)
In Exercises, find the indefinite integral. 2x (x - 1)² dx
In Exercises, use the properties of logarithms to expand the logarithmic expression. In(x√√x² + 5)
In Exercises, find the indefinite integral. dx (8/1x + 1)g/zx 1
In Exercises, use the properties of logarithms to expand the logarithmic expression. I - Du
In Exercises, find the indefinite integral. (x(x - 2) (x - 1)³ fixt dx
In Exercises, use the properties of logarithms to expand the logarithmic expression. In X 1
In Exercises, use the properties of logarithms to expand the logarithmic expression. 1 In e
In Exercises, use the properties of logarithms to expand the logarithmic expression. In(3e²)
In Exercises, use the properties of logarithms to expand the logarithmic expression. In z(z − 1)²
In Exercises find the indefinite integral by u-substitution. Let u be the denominator of the integrand. 1 Sit's 1 + 2x dx
In Exercises, write the expression as a logarithm of a single quantity. In(x - 2) - In(x + 2)
In Exercises find the indefinite integral by u-substitution. Let u be the denominator of the integrand. 1 + 1
In Exercises, write the expression as a logarithm of a single quantity. 3 ln x + 2 ln y - 4 ln z
In Exercises, find the indefinite integral. S Ꮎ cot de
In Exercises find the indefinite integral by u-substitution. Let u be the denominator of the integrand. √x √x - 3 dx
In Exercises find the indefinite integral by u-substitution. Let u be the denominator of the integrand. X 3√x - 1 dx
In Exercises, write the expression as a logarithm of a single quantity. [2 In(x + 3) + In x - In(x² - 1)]
In Exercises, find the indefinite integral. Jese csc 2x dx
In Exercises, find the indefinite integral. tan 50 de
In Exercises, write the expression as a logarithm of a single quantity. 2[In x - In(x + 1) − In(x − 1)] - -
In Exercises (a) Verify that ƒ = g by using a graphing utility to graph ƒ and g in the same viewing window (b) Verify that ƒ = g algebraically. f(x) = In x > 0, g(x) = 2 ln x - In
In Exercises, write the expression as a logarithm of a single quantity. 2 In 3 - -/In(x² + 1)
In Exercises, find the indefinite integral. [ see 1 dx
In Exercises, write the expression as a logarithm of a single quantity. [In(x² + 1) - In(x + 1) − In(x − 1)]
In Exercises, find the indefinite integral. s (cos 30 - 1) de
In Exercises, find the indefinite integral. |(₂ 2 - tan 1/₁ 10
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