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mathematics
calculus early transcendentals

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

The following third- and fourth degree polynomials have a property that makes them relatively easy to graph. Make a complete graph and describe the property.f(x) = x4 + 8x3 - 270x2 + 1
Evaluate the following limits. Use l’Hôpital’s Rule when needed. In x lim Vĩ x-
Evaluate the following limits 1 lim (sin x), x→0+ х
Suppose f has a real root r and Newton’s method is used to approximate r with an initial approximation x0. The basin of attraction of r is the set of initial approximations that produce a sequence that converges to r. Points near r are often in the basin of attraction of r—but not always.
Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x) dx. f(x) = tan x
Determine the following indefinite integrals. Check your work by differentiation. 6. :dx V25 – x?
Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem.f(x) = xe-x
a. Find the critical points of f on the given interval.b. Determine the absolute extreme values of f on the given interval when they exist.c. Use a graphing utility to confirm your conclusions.f(x) = 4x3/3 + 5x2 - 6x on [-4, 1]
An observer stands 20 m from the bottom of a Ferris wheel on a line that is perpendicular to the face of the wheel, with her eyes at the level of the bottom of the wheel. The wheel revolves at a rate of π rad/min, and the observer’s line of sight with a specific seat on the Ferris wheel makes an
Use the graphs of f' and f" to find the critical points and inflection points of f, the intervals on which f is increasing and decreasing, and the intervals of concavity. Then graph f assuming f(0) = 0.
Evaluate the following limits. Use l’Hôpital’s Rule when needed. lim csc x sin х х—0
Evaluate the following limits :)- т х) sec x lim х—п/2 2
Let f(x) = ax(1 - x), where a is a real number and 0 ≤ x ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x. a. Without using a calculator, find the values of a, with 0 < a ≤ 4, such that f has a fixed point. Give the fixed point in terms of a.b.
Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x) dx. f(x) = sin-1 x
Determine the following indefinite integrals. Check your work by differentiation.∫(2et + 2√t) dt
a. Locate the critical points of f.b. Use the First Derivative Test to locate the local maximum and minimum values.c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).f(x) = tan-1 x - x3 on [-1, 1]
a. Find the critical points of f on the given interval.b. Determine the absolute extreme values of f on the given interval when they exist.c. Use a graphing utility to confirm your conclusions.f(x) = |2x - x2| on [-2, 3]
A searchlight is 100 m from the nearest point on a straight highway (see figure). As it rotates, the searchlight casts a horizontal beam that intersects the highway in a point. If the light revolves at a rate of π/6 rad/s, find the rate at which the beam sweeps along the highway as a function of
Use the graphs of f' and f" to find the critical points and inflection points of f, the intervals on which f is increasing and decreasing, and the intervals of concavity. Then graph f assuming f(0) = 0.
Approximate the area of the region bounded by the graph of f(t) = cos (t/2) and the t-axis on [0, π] with n = 4 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure). УА 1 f(t) = cos (t/2) 3п 브2
Evaluate the following integrals. (1 — соs' 30) dө
Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found. Г-9) dx .2 1/2
Use a change of variables to find the following indefinite integrals. Check your work by differentiating.Let u = x - 2. dx
Use a change of variables to find the following indefinite integrals. Check your work by differentiating. dx .2 1 + 4x²
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. V16 – x² dx
Evaluate the following integrals. .3 x² + 2x – 2 dx x³ + 3x? – 6x бх
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · xn.c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.d. Calculate the midpoint
Evaluate the following integrals using the Fundamental Theorem of Calculus. 4x dx
Use a change of variables to find the following indefinite integrals. Check your work by differentiating. 3 dy 1 + 25y2
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. .3 4 – (x – 1)² dx
Evaluate the following integrals. cln 2 et 1 + e2x
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · xn.c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.d. Calculate the midpoint
Evaluate the following integrals using the Fundamental Theorem of Calculus. (3x? + 2х). 2x) dx
Use a change of variables to find the following indefinite integrals. Check your work by differentiating. dx, x > xV4x² – 1 -IN
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. [5 if x < 2 f(x) dx, where f(x) = if x > 2 Зх — 1
Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region.f(x) = 16 - x2 on [-4, 4]
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · xn.c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.d. Calculate the midpoint
Evaluate the following integrals using the Fundamental Theorem of Calculus. (x + Vx) dx
Use a change of variables to find the following indefinite integrals. Check your work by differentiating. 8х + 6 dx 2x2 + 3x
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. 4x if 0 < x < 2 10 g(x) dx, where g(x) if 2 < x < 3 -8x + 16 -8 if x > 3
Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region.f(x) = x3 - x on [-1, 0]
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · xn.c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.d. Calculate the midpoint
Evaluate the following integrals using the Fundamental Theorem of Calculus.
Find the following integrals. х Vх — 4
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 f(x) dx
Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region.f(x) = 2 sin(x/4) on [0, 2π]
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · xn.c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.d. Calculate the midpoint
Evaluate the following integrals using the Fundamental Theorem of Calculus. 2 dx Vx
Find the following integrals. y? dy (y + 1)4
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 f(x) dx
Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region.f(x) = 1/(x2 + 1) on [-1, √3]
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · xn.c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.d. Calculate the midpoint
Evaluate the following integrals using the Fundamental Theorem of Calculus. 2 + Vi dt J 4
Find the following integrals. х dx (x + 4
Find (i) The net area. (ii) The area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region.f(x) = x4 - x2 on [-1, 1]
Evaluate the left and right Riemann sums for f over the given interval for the given value of n.n = 4; [0, 2] 0.5 1.5 f(x) 5 3
Evaluate the following integrals using the Fundamental Theorem of Calculus. Гое 2 (х* — 4) dx (x² -2
Find the following integrals. e* x- dx e* + e*
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 f(x) dx
Find (i) The net area. (ii) The area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region.f(x) = x2 - x on [0, 3]
Evaluate the left and right Riemann sums for f over the given interval for the given value of n.n = 8; [1, 5] 1.5 2.5 3 3.5 4.5 f(x) 3 2 3 2. 4-
Evaluate the following integrals using the Fundamental Theorem of Calculus. In 8 e* dx
Find the following integrals.∫ x 3√2x + 1 dx
The following functions are positive and 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.c. Use the sketch in part (a) to show
Use a change of variables to find the following indefinite integrals. Check your work by differentiating. 2x2 dx V1 – 4x³ .3
Consider the following functions f and real numbers a (see figure).a. Find and graph the area function b. Verify that A'(x) = f(x).f(t) = t + 5, a = -5 A(x) = SÄf(1) dt.
Evaluate the following integrals.∫ (9x8 - 7x6) dx
Complete the following steps for the given function, interval, and value of n.a. Sketch the graph of the function on the given interval.b. Calculate Δx and the grid points x0, x1, · · · · xn.c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and
The following functions are positive and 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.c. Use the sketch in part (a) to show
Use a change of variables to find the following indefinite integrals. Check your work by differentiating.∫ xex2 dx
Let f(t) = 2t - 2 and consider the two area functions and F(x) = a. Evaluate A(2) and A(3). Then use geometry to find an expression for A(x), for x ≥1.b. Evaluate F(5) and F(6). Then use geometry to find an expression for F(x), for x ≥ 4.c. Show that A(x) - F(x) is a constant and that
Evaluate the following integrals. (4x21 – 2x16 + 1) dx
Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n. on [1,5]; n = 4 f(x) УА УА f(x) = 1/x f(x) = 1/x 1 1 2 3 4 2 3 4.
The following functions are positive and 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.c. Use the sketch in part (a) to show
Use a change of variables to find the following indefinite integrals. Check your work by differentiating.∫ 2x(x2 - 1)99 dx
Let f(t) = t and consider the two area functionsand a. Evaluate A(2) and A(4). Then use geometry to find an expression for A(x), for x ≥ 0.b. Evaluate F(4) and F(6). Then use geometry to find an expression for F(x), for x ≥ 2.c. Show that A(x) - F(x) is a constant and that A'(x) = F'(x) = f(x).
Evaluate the following integrals. (x + 1)³ dx
Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.f(x) = x + 1 on [1, 6]; n = 5 УА У f(x) = x + 1 f(x) = x + 1 6. 6. 4 2 1 5 6 x 1 3 4 1 3 4 5 3. 3,
The following functions are positive and 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.c. Use the sketch in part (a) to show
Use the given substitution to find the following indefinite integrals. Check your answer by differentiating.∫ (6x + 1) √3x2 + x dx, u = 3x2 + x
Consider the following functions f and real numbers a (see figure). a. Find and graph the area functionb. Verify that A'(x) = f(x)f(t) = 2, a = -3 A(x) = Säf(1) dt for f. У y = f(t) А(х) — area х
Evaluate the following integrals.∫ cos 3x dx
The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. t +3 (m/s), for 0 st
The following functions are positive and 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.c. Use the sketch in part (a) to show
Use the given substitution to find the following indefinite integrals. Check your answer by differentiating.∫ sin3 x cos x dx, u = sin x
Consider the following functions f and real numbers a (see figure). a. Find and graph the area functionb. Verify that A'(x) = f(x)f(t) = 5, a = -5 A(x) = Säf(1) dt for f. У y = f(t) А(х) — area х
Evaluate the following integrals. •2 (3x* – 2x + 1) dx -2
The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.v = 4√t + 1
The following functions are positive and 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.c. Use the sketch in part (a) to show
Use the given substitution to find the following indefinite integrals. Check your answer by differentiating.∫ 8x cos (4x2 + 3)dx, u = 4x2 + 3
Consider the following functions f and real numbers a (see figure). a. Find and graph the area functionb. Verify that A'(x) = f(x)f(t) = 10, a = 4 A(x) = Säf(1) dt for f. У y = f(t) А(х) — area х
Use geometry to find the area A(x) that is bounded by the graph of f(t) = 2t - 4 and the t-axis between the point (2, 0) and the variable point (x, 0), where x ≥ 2. Verify that A'(x) = f(x).
The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.v = t2/2 + 4 (ft/s),
Use the given substitution to find the following indefinite integrals. Check your answer by differentiating.∫ 2x (x2 + 1)4 dx, u = x2 + 1
Evaluate the following limit by identifying the integral that it represents: 5 4k lim 2 |n→* k=1` п
Consider the following functions f and real numbers a (see figure). a. Find and graph the area functionb. Verify that A'(x) = f(x)f(t) = 5, a = 0 A(x) = Säf(1) dt for f. У y = f(t) А(х) — area х
The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. (m/s), for 0
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) = sin 2x on [π/2, π]
Find an antiderivative of the following functions by trial and error. Check your answer by differentiating.f(x) = cos (2x + 5)
The graph of f is shown in the figure. Let be two area functions for f. Evaluate the following area functions.a. A(2)b. F(5) c. A(0)d. F(8)e. A(8)f. A(5)g. F(2) A(x) = Sf(t) dt and F(x) = [;f(t) di y = f(t) Area = 8 2 + 3 5 -2 Area = 5 Area = 11

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