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study help
mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Evaluate the integral using the methods covered in the text so far. ST 7x dx
Find the local extreme values in the domain {x : x > 0} and use the Second Derivative Test to determine whether these values are local minima or maxima.g(x) = x ln x
Evaluate the integral using the methods covered in the text so far. Sea-12 e9-12t dt
By a fact from algebra, if ƒ, g are polynomials such that ƒ(a) =g(a) = 0, then there are polynomials ƒ1, g1 such that ƒ(x) = (x − a) ƒ1(x), g(x) = (x − a)g1(x)Use this to verify L’Hôpital’s Rule directly for lim f(x)/g(x). ·a
Find the local extreme values in the domain {x : x > 0} and use the Second Derivative Test to determine whether these values are local minima or maxima. g(x)= = In x X
Use L’Hôpital’s Rule to evaluate and check your answers numerically: (a) lim x→0+ sin x X 1/x² (b) lim - x=0\sin’x x²
Evaluate the integral using the methods covered in the text so far. Ss sec² 0 tan² e de
Evaluate the integral using the methods covered in the text so far. cos(In t) dt t S
Find the area between y = ex and y = e2x over [0, 1].
Find the local extreme values in the domain {x : x > 0} and use the Second Derivative Test to determine whether these values are local minima or maxima.g(x) = x − ln x
Find the local extreme values and points of inflection, and sketch the graph of y = ƒ(x) over the interval [1, 4]. f(x) = 10 ln x .x²
Evaluate the integral using the methods covered in the text so far. tdt √7-1²
Find the area between y = ex and y = e−x over [0, 2].
Evaluate the integral using the methods covered in the text so far. S 25 2x e4x dx
Find the area bounded by y = e2, y = ex, and x = 0.
Evaluate the integral using the methods covered in the text so far. (3x + 2) dx 1² +4 Ss
Find the local extreme values and points of inflection, and sketch the graph of y = ƒ(x) over the interval [1, 4].ƒ(x) = x2 − 8 ln x
Find the volume obtained by revolving y = ex about the x-axis for 0 ≤ x ≤ 1.
Wind engineers have found that wind speed v (in m/s) at a given location follows a Rayleigh distribution of the typeThis means that the probability that v lies between a and b is equal to the shaded area in Figure 8.(a) Show that the probability that v ∈ [0, b] is 1 − e−b2/64.(b) Calculate
Evaluate the indefinite integral, using substitution if necessary. 7 dx X
Evaluate the integral using the methods covered in the text so far. Stan tan(4x + 1) dx
Evaluate the indefinite integral, using substitution if necessary. S dx x + 7
The function ƒ(x) = ex satisfies ƒ(x) = ƒ(x). Show that if g is another function satisfying g'(x) = g(x), then g(x) = Cex for some constant C. Compute the derivative of g(x)e−x.
Evaluate the integral using the methods covered in the text so far. S dx √1-16x²
Evaluate the indefinite integral, using substitution if necessary. dx 2x + 4 S
Prove that ƒ(x) = ex is not a polynomial function. Differentiation lowers the degree of a polynomial by 1.
Recall the following property of integrals: If ƒ(t) ≥ g(t) for all t ≥ 0, then for all x ≥ 0, f* f(1)dt = f*g(1) dt z The inequality et ≥ 1 holds for t≥ 0 because e > 1. Use (4) to prove that e 21+x for x ≥ 0 Then prove, by successive integration, the following inequalities (for x ≥
Evaluate the integral using the methods covered in the text so far. Sen e¹ √e¹ + 1 dt
Generalize Exercise 86; that is, use induction (if you are familiar with this method of proof) to prove that for all n ≥ 0,Exercise 86Recall the following property of integrals: If ƒ(t) ≥ g(t) for all t ≥ 0, then for all x ≥ 0, 1 ex > 1 + x + - x2 > 2 + + ... + 1 6 1 - n! (x > 0)
Evaluate the indefinite integral, using substitution if necessary. dx 9x - 3 S3
Use Exercise 86 to show that ex/x2 ≥ x/6 and conclude that Then use Exercise 87 to prove more generally thatData From Exercise 86 Recall the following property of integrals: If ƒ(t) ≥ g(t) for all t ≥ 0, then for all x ≥ 0,Data From Exercise 87Generalize Exercise 86; that is, use
Evaluate the integral using the methods covered in the text so far. (ex. - 4x) dx
Evaluate the indefinite integral, using substitution if necessary. tdt 2² +4 S
Evaluate the integral using the methods covered in the text so far. (7-e¹0x) dx
Evaluate the indefinite integral, using substitution if necessary. x² dx X 1³ +2
Evaluate the integral using the methods covered in the text so far. e²x - e4x ex dx
Calculate the first three derivatives of ƒ(x) = xex. Then guess the formula for ƒ(n)(x) (use induction to prove it if you are familiar with this method of proof).
Evaluate the indefinite integral, using substitution if necessary. (3x - 1) dx 9 - 2x + 3.1² Sz
Evaluate the integral using the methods covered in the text so far. dx x √25x² - 1 S
Consider the equation ex = λx, where λ is a constant.(a) For which λ does it have a unique solution? For intuition, draw a graph of y = ex and the line y = λx.(b) For which λ does it have at least one solution?
Evaluate the indefinite integral, using substitution if necessary. S₁ tan(4x + 1) dx
Evaluate the integral using the methods covered in the text so far. dx (4x 1) In(8x - 2) S
Evaluate the indefinite integral, using substitution if necessary. dx x ln x Sz
Evaluate the integral using the methods covered in the text so far. Serce². e¹(e²x + 1)³ dx
Evaluate the indefinite integral, using substitution if necessary. dx (4x 1) In(8.x - 2) S
Evaluate the integral using the methods covered in the text so far. S dx x(In x)5
Evaluate the indefinite integral, using substitution if necessary. In(In x) x ln x St dx
Evaluate the indefinite integral, using substitution if necessary. So cotx ln(sin x) dx
Evaluate the integral using the methods covered in the text so far. S X x² dx 13 +2
Evaluate the indefinite integral, using substitution if necessary. 3* dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. 4x - 12 lim x3 x2 – 5r+6
Evaluate the integral using the methods covered in the text so far. (3x - 1) dx 9- 2x + 3x²
Evaluate the indefinite integral, using substitution if necessary. Sx x3x² dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. x³ + 2x²-x-2 lim x2 14 +2.1³4x- 8.
Evaluate the integral using the methods covered in the text so far. cot x dx
Evaluate the indefinite integral, using substitution if necessary. cos cos x 3 sinx dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. lim x¹/² In x X→0+
Evaluate the integral using the methods covered in the text so far. Sz COS X 2 sin x + 3 dx
Evaluate the indefinite integral, using substitution if necessary. 3x+2 S (12) dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. lim In(e¹ + 1) t
Evaluate the integral using the methods covered in the text so far. 4 ln x + 5 X dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. 2 sin lim 0-0 sin sin 20 cos 0
Evaluate the integral using the methods covered in the text so far. f(s (sec 0 tan 0)5sec de 8
Verify that L’Hôpital’s Rule applies and evaluate the limit. lim X-0 √4+x-2v1 + x 1²
Evaluate the integral using the methods covered in the text so far. Sr. x3x² dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. lim 1-00 In(t + 2) log₂ t
Evaluate the integral using the methods covered in the text so far. Sime In(In x) x ln x dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. ex lim x-0e - 1 X
Evaluate the integral using the methods covered in the text so far. Sco cot x In(sin x) dx
Verify that L’Hôpital’s Rule applies and evaluate the limit. lim y→0 siny-y
Evaluate the integral using the methods covered in the text so far. J t dt V1 √1 - 14
Use Figure 12 to prove 1 1 So VI-P dt = XVI-X² + sin ¹ x
Verify that L’Hôpital’s Rule applies and evaluate the limit. V1-x² lim x-1-cos-¹x
Find a good numerical approximation to the coordinates of the point on the graph of y = ln x − x closest to the origin (Figure 8). 0.5- 1 + 2 3 4 +x 5
Verify that L’Hôpital’s Rule applies and evaluate the limit. sinh(x2) lim x-0 cosh x - 1
Verify that L’Hôpital’s Rule applies and evaluate the limit. lim tanh x – sinh sin x - x
The function ƒ(x) = x1/x has an absolute maximum value over x > 0. Find it and where it occurs.
Prove:∫sin−1 t dt = √1 − t2 + t sin−1 t + C
Use the formula (ln ƒ(x))'ƒ'(x)/ ƒ(x) to show that ln x and ln(2x) have the same derivative. Is there a simpler explanation of this result?
A cylindrical tank of radius R and length L lying horizontally as in Figure 13 is filled with oil to height h.(a) Show that the volume V(h) of oil in the tank as a function of height h is(b) Show that (c) Suppose that R = 2 m and L = 12 m, and that the tank is filled at a constant rate of 1.5
Explain why L’Hôpital’s Rule gives no information about Evaluate the limit by another method. 2.x - sin x lim x+ 3x + cos2x
Let ƒ be a differentiable function with inverse g such that ƒ(0) = 0 and ƒ'(0) ≠ 0. Prove that f(x) lim = f'(0)² x→0 g(x)
(a) Show that if ƒ and g are differentiable, then(b) Give a new proof of the Product Rule by observing that the left-hand side of Eq. (8) is equal to (ƒ(x)g(x))'/ƒ'(x)g(x). d dx In(f(x)g(x)) = f'(x) g'(x) + f(x) g(x)
In this exercise, we prove that for all x > 0, X- 2 < ln(1 + x) ≤ x
Develop an elegant approach to the exponential and logarithm functions. Define a function G for x > 0:This exercise proceeds as if we didn’t know that G(x) = ln x and shows directly that G has all the basic properties of the logarithm. Prove the following statements. G(x) = S² JI 1 - dt
Prove the formula loga b logb a = 1 for all positive numbers a, b with a ≠ 1 and b ≠ 1.
LetProve that F(x) and cosh−1 x differ by a constant by showing that they have the same derivative. Then prove they are equal by evaluating both at x = 1. 25₁² F(x)=x√x²-1-2 √₁²2 - 1 dt
Let gd(y) = tan−1(sinh y) be the so-called gudermannian, which arises in cartography. In a map of the earth constructed by Mercator projection, points located y radial units from the equator correspond to points on the globe of latitude gd(y). Prove that d dy -gd(y) = sechy.
Prove in two ways that the numbers m(a) satisfy m(ab) = m(a) + m(b)(a) First method: Use the limit definition of m(b) and(b) Second method: Apply the Product Rule to axbx = (ab)x. (ab)" - 1 = B² (r^²= -1) + b² = 1 - Bh h h
Let ƒ(y) = 2 tan−1(ey) − π/2. Prove that gd(y) = ƒ(y). Show that gd(y) = ƒ'(y) and ƒ (0) = gd(0).
Evaluate the integral using the methods covered in the text so far. (x + 5) dx √4x²
Show that t(y) = sinh−1(tan y) is the inverse of gd(y) for 0 ≤ y < π/2.
Evaluate the indefinite integral, using substitution if necessary. Scot cotx dx
Evaluate the integral using the methods covered in the text so far. (t+1) √t + 1 dt
Evaluate the indefinite integral, using substitution if necessary. COS X £2500+ 2 sin x + 3 dx
Evaluate the integral using the methods covered in the text so far. Se e cos(e) dx
Evaluate the indefinite integral, using substitution if necessary. In x fint dx
Evaluate the integral using the methods covered in the text so far. ex √ex +1 S dx
Evaluate the integral using the methods covered in the text so far. S dx √9-16.x²
Evaluate the indefinite integral, using substitution if necessary. 4 ln x + 5 dx X
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