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mathematics
calculus early transcendentals

Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions

If necessary, use two or more substitutions to find the following integrals. V1 - Vī dx х
Another change of variables that can be interpreted geometrically is the scaling u = cx, where c is a real number. Prove and interpret the fact thatDraw a picture to illustrate this change of variables in the case that f(x) = sin x, a = 0, b = π, and c = 1/2. Lлоза-Г пом. ebc F(сх) dx ac
Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals.(u = p√x + a and u = x + a) xVx + a dx; a > 0
Evaluate the following integrals in which the function f is unspecified. Note that f(p) is the pth derivative of f and fp is the pth power of f. Assume f and its derivatives are continuous for all real numbers.∫ (5f3(x) + 7f2(x) + f(x)) f'(x) dx
The area of the shaded region under the curveon the interval [4, 9] in (a) equals the area of the shaded region under the curve y = x2 on the interval [1, 2] in (b). Without computing areas, explain why. (Vx – 1)ª у з 2Vx y, УА УА y = x² 4 3+ (Vĩ – 1)2 y = 2 - 2 2Vx 1 4 х (a) (b) 3,
a. Multiply the numerator and denominator of sec x by sec x + tan x; then use a change of variables to show that∫ sec x dx = ln |sec x + tan x| + C.b. Use a change of variables to show that∫ csc x dx = -ln |csc x + cot x| + C.
Use a change of variables to evaluate the following definite integrals. In 4 et dx 3 + 2e*
Use a change of variables to evaluate the following definite integrals. •1/V3 4 dx 9x2 + 1 J 1/3
Use a change of variables to evaluate the following definite integrals. 1/4 х dx /1 – 16x²
Use a change of variables to evaluate the following definite integrals. n4 dx x2 + 1 0.
Use a change of variables to evaluate the following definite integrals. .3 v2 + 1 dv v³ + 3v + 4
Use a change of variables to evaluate the following definite integrals. r2/5 dx 2/(5V3) xV25x²2 – 1
Use a change of variables to evaluate the following definite integrals. 7/4 sin 0 - do 3 cos 0
Use a change of variables to evaluate the following definite integrals. 7/2 cos x dx sin x T/4
Use a change of variables to evaluate the following definite integrals. dp V9 + p²
Use a change of variables to evaluate the following definite integrals. хРеt+1 dx -1
Use a change of variables to evaluate the following definite integrals. 7/4 sin x dx. cos“ x
Use a change of variables to evaluate the following definite integrals. 7/2 sin? 0 cos 0 do
Use a change of variables to evaluate the following definite integrals. 2x dx (x² + 1)²
Use a change of variables to evaluate the following definite integrals. 2x(4 — х?) dx
Find the following integrals.∫ (z + 1)√3z + 2 dz
Use a change of variables to find the following indefinite integrals. Check your work by differentiating.∫ sin10 θ cos θ dθ
Use a change of variables to find the following indefinite integrals. Check your work by differentiating.∫ x3(x4 + 16)6 dx
Use a change of variables to find the following indefinite integrals. Check your work by differentiating. по dx 10х — 3
Graph the function f(x) = 8 + 2x - x2 and determine the values of a and b that maximize the value of the integral (8 + 2x – x²) dx.
What value of b > -1 maximizes the integral x² (3 – x) dx? -1
Simplify the given expressions. e' dt dx х
Simplify the given expressions. COs x cos. (1ª + 6) dt dx Jo
Simplify the given expressions. dt 12 + 4 dx
Find the area of the region bounded by the graph of f and the x-axis on the given interval.f(x) = x2 - 25 on [2, 4]
Find (i) The net area.(ii) The area of the following regions. Graph the function and indicate the region in question.The region above the x-axis bounded by y = 4 - x2.
Find (i) The net area.(ii) The area of the following regions. Graph the function and indicate the region in question.The region bounded by y = x1/2 and the x-axis between x = 1 and x = 4.
Evaluate the following integrals using the Fundamental Theorem of Calculus. TT/8 8 csc2 4x dx 7/16
Evaluate the following integrals using the Fundamental Theorem of Calculus. dx 1 + x?
Evaluate the following integrals using the Fundamental Theorem of Calculus. 10e2* dx ,2x
Evaluate the following integrals using the Fundamental Theorem of Calculus. 7/8 cos 2x dx
Evaluate the following integrals using the Fundamental Theorem of Calculus. VI dx .3
Evaluate the following integrals using the Fundamental Theorem of Calculus. 3 – dt
Evaluate the following integrals using the Fundamental Theorem of Calculus. 7/2 (cos x – 1) dx -T/2
Evaluate the following integrals using the Fundamental Theorem of Calculus. Го | (1 - x)(x – 4).
Evaluate the following integrals using the Fundamental Theorem of Calculus. п (1 – sin x) dx Г0
Evaluate the following integrals using the Fundamental Theorem of Calculus. -1 xp -2 -2
Evaluate the following integrals using the Fundamental Theorem of Calculus. •1/2 dx (1 – x?
Evaluate the following integrals using the Fundamental Theorem of Calculus. -п/4 TT sec? @ dө
Evaluate the following integrals using the Fundamental Theorem of Calculus. x(x – 2)(x – 4) dx
Evaluate the following integrals using the Fundamental Theorem of Calculus. (x-3 – 8) dx / 1/2
Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure. -7п/4 (sin x + cos r) dx 7/4 УА y = sin x + cos x 3т 7п х 4 -2-
Determine whether the following statements are true and give an explanation or counterexample.a. If f is a constant function on the interval [a, b], then the right and left Riemann sums give the exact value offor any positive integer n.b. If f is a linear function on the interval [a, b], then a
Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. 4х3 dx
Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. Г - п aк | (x² – 1) dx
Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. (г2 1) dx
Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. .7 (4х + 6) dx 3
Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. Гo (1 — х) dx
Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. Г (2х + 1) dx
Use the value of the first integral I to evaluate the two given integrals.a.b. CT/2 (2 sin 0 – cos 0) do /2 т (2 sin @ — cos ®) dө
Use the value of the first integral I to evaluate the two given integrals.a. b. 3 4 (x³ – 2x) dx = - I = (4х — 2r3) dx 2x³) dx
Suppose f(x) ≥ 0 on [0, 2], f(x) ≤ 0 on [2, 5],Evaluate the following integrals.a. b. c. d. S(x) dx = 6, and Si 5(x) dx = -8. .5 S(х) dx
Suppose and Evaluate the following integrals.a.b.c.d. SS4) dx = 2. fi[(x) dx = -5. Sis(x) dx = 1. 3
SupposeandEvaluate the following integrals.a.b.c.d. Sis«) dx = 8 SiS(«) dx = = 5.
Use only the fact that and the definitions and properties of integrals to evaluate the following integrals, if possible.a.b.c.d. J 3x(4 — х) dx%3D 32 | 3x(4 – x) dx 4)
The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, π - 1, π + 1, and 2π - 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. УА Area = 1 Area = T + 1 y = x
The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, π - 1, π + 1, and 2π - 1, respectively. Use this information to evaluate the following integrals. УА Area = 1 Area = T + 1 y = x sin x R2 R1 3п х п 2т TT 2 R3 -2 R4 Area =
The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, π - 1, π + 1, and 2π - 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. УА Area = 1 Area = T + 1 y = x
The accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, π - 1, π + 1, and 2π - 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. УА Area = 1 Area = T + 1 y = x
The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. УА y = f(x) 16 11 а 5 /Ь х Г Г(x) dx
Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. (8 — 2х) dx
Consider the following limits of Riemann sums for a function f on [a, b]. Identify f and express the limit as a definite integral. п lim Ι x-1Δxχ on [-2, 2] Δ-0 Κ1 k=1
Consider the following limits of Riemann sums for a function f on [a, b]. Identify f and express the limit as a definite integral. lim Exi In xAxg on [1, 2] A→0 k=1
The following functions are negative on the given interval.a. Sketch the function on the given interval.b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.f(x) = -4 - x3 on [3, 7]
The following functions are negative on the given interval.a. Sketch the function on the given interval.b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.f(x) = -2x - 1 on [0, 4]
Leta. Find the intervals on which f is increasing and the intervals on which f is decreasing.b. Find the intervals on which f is concave up and the intervals on which f is concave down.c. For what values of x does f have local minima? Local maxima?d. Where are the inflection points of f ? = L,(t
Evaluate the following integrals. п/4 cos? 80 do
Evaluate the following integrals. т de ө + sin?( 0 6.
Evaluate the following integrals. TT / cos?. cos? x dx
Determine whether the following statements are true and give an explanation or counterexample.a. Consider the linear function f(x) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3, 6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then
Find (i) The net area.(ii) The area of the following regions. Graph the function and indicate the region in question.The region below the x-axis bounded by y = x4 - 16.
Evaluate the following integrals.∫ sin2 x dx
Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration.b. Calculate Δx and the grid points x0, x1, · · · · , xn, assuming a regular partition.c. Calculate the left and right Riemann sums for the
Consider the graph of the continuous function f in the figure and letAssume the graph consists of a line segment from (0, -2) to (2, 2) and two quarter circles of radius 2.a. Evaluate F(2), F(-2), and F(4).b. Evaluate G(-2), G(0), and G(4).c. Explain why there is a constant C such that F(x) = G(x)
Let f(x) = √1 - x2.a. Show that the graph of f is the upper half of a circle of radius 1 centered at the origin.b. Estimate the area between the graph of f and the x-axis on the interval [-1, 1] using a midpoint Riemann sum with n = 25.c. Repeat part (b) using n = 75 rectangles. d. What
Find (i) The net area.(ii) The area of the following regions. Graph the function and indicate the region in question.The region bounded by y = 6 cos x and the x-axis between x = -π/2 and x = π
Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration.b. Calculate Δx and the grid points x0, x1, · · · · , xn, assuming a regular partition.c. Calculate the left and right Riemann sums for the
Consider the functionand its graph shown below.Leta. Evaluate F(-2) and F(2).b. Use the Fundamental Theorem to find an expression for F'(x), for -2 ≤ x < 0.c. Use the Fundamental Theorem to find an expression for F'(x), for 0 ≤ x ≤ 2.d. Evaluate F'(-1) and F'(1). Interpret these values.e.
Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.The right Riemann sum for f(x) = x + 1 on [0, 4] with n = 50
Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration.b. Calculate Δx and the grid points x0, x1, · · · · , xn, assuming a regular partition.c. Calculate the left and right Riemann sums for the
Without evaluating the integrals, explain why the following statement is true for positive integers n:
Fill in the blanks with right or midpoint, an interval, and a value of n. In some cases, more than one answer may work.is a ______ Riemann sum for f on the interval [___, ___] with n = ______. |Ef( 1.5 + Σ 2 k=1
Evaluate the following integrals. sin - dx .2 х
Evaluate the following integrals. [(tan-x)5 (tan x) -dp- 1 + x?
Evaluate the following integrals. dx -1 ()(1 + x²) (tan
Evaluate the following integrals. sin-1 /1 - х2
Evaluate the following integrals. e* e t dx e* + e¯*
Let a > 0 be a real number and consider the family of functions f(x) = sin ax on the interval [0, π/a]. a. Graph f, for a = 1, 2, 3.b. Let g(a) be the area of the region bounded by the graph of f and the x-axis on the interval [0, π/a]. Graph g for 0 < a < ∞. Is g an increasing
Explain why if a function u satisfies the equationthen it also satisfies the equation u'(x) + 2u(x) = 0. Is it true that if u satisfies the second equation, then it satisfies the first equation? u(x) + 2/,u(1) dt = 10,
Consider the function f(t) = t2 - 5t + 4 and the area functiona. Graph f on the interval [0, 6].b. Compute and graph A on the interval [0, 6].c. Show that the local extrema of A occur at the zeros of f.d. Give a geometrical and analytical explanation for the observation in part (c).e. Find the
Sketch a graph of f(t) = et on an arbitrary interval [a, b]. Use the graph and compare areas of regions to prove that eb – ea eª + eb ela+b)/2 < b – a 2
Evaluate where 0 < a < b, using the definition of the definite integral and the following steps.a. Assume {x0, x1, · · · ·, xn} is a partition of [a, b] with Show that b. Show that c. Simplify the general Riemann sum forusing d. Conclude that dx .2' Axg = xg – X–1,
Evaluate the following definite integrals using the Fundamental Theorem of Calculus. In 2 e* dx
Evaluate the following definite integrals using the Fundamental Theorem of Calculus.
Evaluate the following definite integrals using the Fundamental Theorem of Calculus. ds

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