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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Compute the given approximation to the integral. Si dx x³ + 1' T6
Let F(x) = ∫x2 dt/ln t and G(x) = x/ln xVerify that L’Hôpital’s Rule applies to the limit and evaluate L. F(x) L = lim x-00 G(x)
Compute the given approximation to the integral. So e-² dx, S4
Compute the given approximation to the integral. Sº 5 cos(x²) dx, S8
An improper integral I = ∫∞a ƒ(x) dx is called absolutely convergent if ∫∞a |ƒ(x)| dx converges. It can be shown that if I is absolutely convergent, then it is convergent.Show that
The following table gives the area A(h) of a horizontal cross section of a pond at depth h. Use the Trapezoidal Rule to estimate the volume V of the pond (Figure 1). h (ft) 0 2 4 6 8 A(h)
Suppose that the second derivative of the function A in Exercise 99 satisfies |A"(h)| ≤ 1.5. Use the Error Bound to find the maximum possible error in your estimate of the volume V of the pond.Data
An improper integral I = ∫∞a ƒ(x) dx is called absolutely convergent if ∫∞a |ƒ(x)| dx converges. It can be shown that if I is absolutely convergent, then it is convergent.Show that
An improper integral I = ∫∞a ƒ(x) dx is called absolutely convergent if ∫∞a |ƒ(x)| dx converges. It can be shown that if I is absolutely convergent, then it is convergent.Let ƒ(x) = sin
The gamma function, which plays an important role in advanced applications, is defined for n ≥ 1 by(a) Show that the integral defining Γ(n) converges for n ≥ 1 (it actually converges for all n
Let ƒ(x) = sin(x3). Find a bound for the errorFind a bound K2 for |ƒ"(x)| by plotting ƒ" with a graphing utility. T24- - T/2 f(x) dx
For which value of m is the following statement correct? If ƒ(2) = 3 and ƒ(4) = 9, and f is differentiable, then ƒ has a tangent line of slope m.
Estimate using the Linear Approximation or linearization, and use a calculator to estimate the error.8.11/3 − 2
True or False? The Linear Approximation says that the vertical change in the graph is approximately equal to the vertical change in the tangent line.
Find the dimensions x and y of the rectangle of maximum area that can be formed using 3 m of wire.(a) What is the constraint equation relating x and y?(b) Find a formula for the area in terms of x
If ƒ is concave up, then ƒ' is (choose one)(a) Increasing (b) Decreasing
What is the definition of a critical point?
Match the graphs in Figure 13 with the description:(a) ƒ"(x)(b) ƒ"(x) goes from + to −.(c) ƒ"(x) > 0 for all x. (d) ƒ"(x) goes from − to +. (A) (B) (C) 2 (D)
Sketch an arc where ƒ' and ƒ'' have the sign combination ++. Do the same for −+.
If the sign combination of ƒ' and ƒ'', changes from ++ to +− at x = c, then (choose the correct answer)(a) ƒ(c) is a local min. (b) ƒ (c) is a local max.(c) (c, ƒ (c)) is a point of
The second derivative of the function C(x) = (x − 4)−1 is ƒ" (x) = 2(x − 4)−3. Although ƒ:(x) changes sign at x = 4, ƒ does not have a point of inflection at x = 4. Why not?
Evaluate the integral using FTC I. -3 np-n "J
Evaluate the integral using FTC I. Sa 10 (12x5 + 3x² - 4x) dx
Solve the differential equation with the given initial condition. dy = 3t² + cost, dt + cost, y(0) = 12
Calculate the integral, assuming that∫50(ƒ(x) – x)
Let V be the volume of a pyramid of height 20 whose base is a square of side 8.(a) Use similar triangles as in Example 1 to find the area of the horizontal cross section at a height y.(b) Calculate V
Find the area of the region enclosed by the graphs of the functions.y = x3 − 2x2 + x, y = x2 − x
Which assumption about fluid velocity did we use to compute the flow rate as an integral?
What does it mean when the integral used to calculate work gives a negative answer?
Compute the work (in joules) required to stretch or compress a spring as indicated, assuming a spring constant of k = 800 N/m.Compressing from equilibrium to 4 cm past equilibrium
(a) Sketch the solid obtained by revolving the region under the graph of ƒ about the x-axis over the given interval, (b) Describe the cross section perpendicular to the x-axis located at x,
Which of the following integrals expresses the volume obtained by rotating the area between y = ƒ(x) and y = g(x) over [a, b] around the x-axis? [Assume ƒ(x) ≥ g(x) ≥ 0.] (a) π * (f(x) =
Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18). y R h
Explain what ∫ba |ƒ(x) − g(x)| dx represents.
Find the area of the region enclosed by the graphs of the functions.x = 4y, x = 24 − 8y, y = 0
Sketch the region between y = sin x and y = cos x over the interval and find its area. 元元|2 42
Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume.ƒ(x) = √x2 + 9, [0, 3]
The average value of ƒ on [1, 4] is 5. Find ∫41 ƒ(x) dx.
Compute the work (in joules) required to stretch or compress a spring as indicated, assuming a spring constant of k = 800 N/m.Stretching from 5 to 15 cm past equilibrium
Find the volume of revolution about the x-axis for the given function and interval. f(x)= 3x-x², [0,3]
Find the volume of the wedge in Figure 19(A) by integrating the area of vertical cross sections. 8 (A) 6 a (B) b
Draw a region that is both vertically simple and horizontally simple.
Find the area of the region enclosed by the graphs of the functions.x = y2 − 9, x = 15 − 2y
Sketch the region between y = sin x and y = cos x over the interval and find its area.[0, π]
Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume. f(x) = x V1 + x³ [1,4]
We investigate nonlinear springs. A spring is linear if it obeys Hooke’s Law, which indicates that the applied force to stretch the spring is F(x) = kx. For a linear spring, F is constant.
Use the method of Examples 2 and 3 to calculate the work against gravity required to build the structure out of a lightweight material of density 600 kg/m3.Example 2Example 3Box of height 3 m and
(a) When rotated about the y-axis, the segment AB generates a disk with radius R = h(y) and the segment CB generates a shell with radius x and height ƒ(x).(b) Based on Figure 11(A) and the
Which of the following quantities is undefined? (a) sin (c) csc -1¹(--) c-¹ ( 1 ) -12 (b) cos ¹(2) (d) csc ¹(2)
Which of the following is equal to d/dx2x? (a) 2x (c) x2x-1 (b) (In 2)2x 1 (d) -2x In 2
State whether each of the following integrals converges or diverges: (a) 00% S³ x-³ dx (c) √ x ² x-2/3 dx (b) (d) S J x-³ dx x-2/3 dx
Is ∫π/20 cot x dx an improper integral? Explain.
Find a value of b > 0 that makes ∫b0 1/x2 − 4 dx an improper integral.
Explain why it is not possible to draw any conclusions about the convergence of dx by comparing with the integral 00 J1 e-x X
Which comparison would show that ∫∞0 dx/x + ex converges?
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. fxs x sin x dx
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. S √1 + x² dx V1
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. 1 + x² 1-x² dx
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. S cos² cos²xsin x dx
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. xpxux xS
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. s √1-x² dx
For each of the following, state what method applies and how one applies it, but do not evaluate the integral. S si sin³ x cos²x dx
For each of the following, find the formula in the integral table at the back of the book that can be applied to find the integral. 3x² dx 5x+2
For each of the following, find the formula in the integral table at the back of the book that can be applied to find the integral. √25 + 16x² x² dx
For each of the following, find the formula in the integral table at the back of the book that can be applied to find the integral. f see sec ³ (4x) dx
For each of the following, find the formula in the integral table at the back of the book that can be applied to find the integral. S x² √x² + 2x +5 dx
Suppose that ∫ ƒ(x) dx = ln x + √x + 1 + C. Can f be a rational function? Explain.
Which of the following are proper rational functions? (a) (c) X X 3 x² + 12 (x + 2)(x + 1)(x - 3) (b) (d) 4 9-x 4x³-7x (x-3)(2x + 5) (9-x)
Which of the following quadratic polynomials are irreducible? To check, complete the square if necessary.(a) x2 + 5 (b) x2 − 5(c) x2 + 4x + 6 (d) x2 + 4x + 2
Let P/Q be a proper rational function where Q(x) factors as a product of distinct linear factors (x − ai). Then(choose the correct answer):(a) Is a sum of logarithmic terms Ai ln(x − ai) for some
Which hyperbolic substitution can be used to evaluate the following integrals? dx √x² + 1 (a) ·S. dx (b) S √² +9 os (c) dx √9x² + 1
Which antiderivative of y = (1 − x2)−1 should we use to evaluate the integral J3 (1-x²)-¹ dx?
Which two of the hyperbolic integration formulas differ from their trigonometric counterparts by a minus sign?
State the trigonometric substitution appropriate to the given integral: (a) (c) √9 - x² dx S√9 [x²(x² + 16) ³/² dx (b) fx²(x²³ - 16 ³1/² dx (d) f(x²-5)-² dx
Is trigonometric substitution needed to evaluate Sx√9- x √9 – x² dx? -
Express sin 2θ in terms of x = sin θ.
Describe the technique used to evaluate Ss sin³ x dx.
Draw a triangle that would be used together with the substitution x = 3 sec θ.
Describe a way of evaluating S si sin x dx.
Are reduction formulas needed to evaluate Ss sin7 x cos² x dx? Why or why not?
Describe a way of evaluating fsin sin x cos xdx.
Which integral requires more work to evaluate?Explain your answer. fsi sin 798 x cos x dx or f sin sin4x cos4 x dx
For each of the following integrals, state whether substitution or Integration by Parts should be used: [xcos(x) dx, xcos xdx, [x²e²dx, xe²³ dx
Compute ∫dx/x2 − 1 in two ways and verify that the answers agree: first via trigonometric substitution and then using the identity 1 x²-1 2 1 1 x-1 1 x + 1
Calculate the integral in terms of the inverse hyperbolic functions. S dx √9x² - 4
Calculate the integral in terms of the inverse hyperbolic functions. dx √4 + x² s
Calculate the integral in terms of the inverse hyperbolic functions. dx V1 + 3x² S
Calculate the integral in terms of the inverse hyperbolic functions. S √x² - 1dx
Calculate the integral in terms of the inverse hyperbolic functions. S x2² dx x² + 1
Calculate the integral in terms of the inverse hyperbolic functions. 1/2 11/2² dx -1/2 1-x²
Calculate the integral in terms of the inverse hyperbolic functions. J4 dx 1-x²
Calculate the integral in terms of the inverse hyperbolic functions. So dx o √1 + x²
Calculate the integral in terms of the inverse hyperbolic functions. -10 dx J2 4x² - 1
Calculate the integral in terms of the inverse hyperbolic functions. -3 dx x√x² + 16
Calculate the integral in terms of the inverse hyperbolic functions. 0.8 dx 0.2 x√1-x²
Calculate the integral in terms of the inverse hyperbolic functions. S √x² - 1dx x²
Calculate the integral in terms of the inverse hyperbolic functions. S dx x√√x + 1
Verify that tanh−1 X = In 2 1 + x 1-x for x < 1.
Evaluate ∫ √x2 − 9 dx in two ways: using trigonometric substitution and using hyperbolic substitution. Then use Exercise 31 to verify that the two answers agree.Data From Exercise 31 Verify
Prove the reduction formula for n ≥ 2: I cos 1 cosh" xdx = cosh"-1 n x sinh x + n-1 ¹=¹ Sco n cosh" 2 xdx

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