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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
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 = et (m/s), for 0
Find an antiderivative of the following functions by trial and error. Check your answer by differentiating.f(x) = √2x + 1
Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. (x³ – x) dx
The graph of f is shown in the figure. Letand be two area functions for f . Evaluate the following area functions.a. A(-2)b. F(8)c. A(4)d. F(4)e. A(8) A(x) = S*, f(t) dt F(x) = SS(t) di 4
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 = 2t + 1 (m/s), for
Find an antiderivative of the following functions by trial and error. Check your answer by differentiating.f(x) = e3x + 1
Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. (3х2 + х) dx + x) dx
Explain why |Saf'(x) dx = f(b) – f(a).
The velocity in ft/s of an object moving along a line is given by v = √10t on the interval 1 ≤ t ≤ 7.a. Divide the time interval [1, 7] into n = 3 subintervals, [1, 3], [3, 5], and [5, 7]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint
If f is continuous on [a, b] andwhat can you conclude about f ? SIS(x) | dx = 0.
Find an antiderivative of the following functions by trial and error. Check your answer by differentiating.f(x) = (x + 1)12
Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. (x2 – 4) dx
Evaluate and where a and b are constants. f(t) dt dx f(t) dt, dx -| f(1)
The velocity in ft/s of an object moving along a line is given by v = 3t2 + 1 on the interval 0 ≤ t ≤ 4.a. Divide the interval [0, 4] into n = 4 subintervals, [0, 1], [1, 2], [2, 3], and [3, 4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the
Use geometry to find a formula forin terms of a. a Sox dx,
What identity is needed to find ∫ sin2 x dx?
Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. | (4x – 2) dx Го
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Determine whether the following statements are true and give an explanation or counterexample. Assume f and f' are continuous functions for all real numbers. a. If
Evaluate the following limits in terms of the parameters a and b, which are positive real numbers. In each case, graph the function for specific values of the parameters to check your results. lim (a* – b*)'/*, a > b > 0 x→0+
Two objects move along the x-axis with position functions x1(t) = 2 sin t and x2(t) = sin (t - π/2). At what times on the interval [0, 2π] are the objects closest to each other and farthest from each other?
Use analytical methods to evaluate the following limits. lim x- In[ cos х х-
Evaluate the following limits. Use l’Hôpital’s Rule when needed. In r100 lim Vĩ
Evaluate the following limits lim (csc (1/x)(e/* – 1))|
A certain kind of differential equation leads to the root-finding problem tan πλ = λ, where the roots λ are called eigenvalues. Find the first three positive eigenvalues of this problem.
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) = 3x3 - 4x
Determine the following indefinite integrals. Check your work by differentiation.∫1/2y dy
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) = √x ln x on (0, ∞)
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) = 2x3 - 15x2 + 24x on [0, 5]
An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 ft above eye level, and the upper edge of the screen is 10 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle θ? 10 ft 3 ft Ө х
Use the derivative f' to determine the x-coordinates of the local maxima and minima of f , and the intervals of increase and decrease. Sketch a possible graph of (f is not unique).f'(x) = x2(x + 2)(x - 1)
Evaluate the following limits. Use l’Hôpital’s Rule when needed. х4 — х3 — 3х2 + 5х — 2 lim х>1 3x2 + 5x x3 + x2 – 5x + 3
Evaluate the following limits lim csc 6x sin 7x
The sinc function, sinc(x) = sin x/x for x ≠ 0, sinc (0) = 1, appears frequently in signal-processing applications.a. Graph the sinc function on [-2π, 2π].b. Locate the first local minimum and the first local maximum of sinc (x), for x > 0.
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) = (4 + x)/(4 - x)
Determine the following indefinite integrals. Check your work by differentiation.∫(csc2 6x dx
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) = x2/x2 - 1 on [-4, 4]
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) = x√2 - x2 on [-√2, √2]
Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?
Use the derivative f' to determine the x-coordinates of the local maxima and minima of f , and the intervals of increase and decrease. Sketch a possible graph of (f is not unique).
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 3 sin 80 lim 0→0 8 sin 30
Evaluate the following limits lim (1 – x) тх 2.
The displacement of a particular object as it bounces vertically up and down on a spring is given by y(t) = 2.5e-t cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure).a. Find the time at which the object first passes the rest position, y =
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) = 2 - a cos x, a constant
Determine the following indefinite integrals. Check your work by differentiation.∫(sec 4θ tan 4θ dθ
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) = x2/3 (x - 5) on [-5, 5]
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) = x2 + cos-1 x on [-1, 1]
a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm3 that minimize the surface area.b. Real problem Compare your answer in part (a) to a real soda can, which has a volume of 354 cm3, a radius of 3.1 cm, and a height of 12.0 cm, to conclude that real
Use the derivative f' to determine the x-coordinates of the local maxima and minima of f , and the intervals of increase and decrease. Sketch a possible graph of (f is not unique).f'(x) = 10 sin 2x on [-2π, 2π]
Evaluate the following limits. Use l’Hôpital’s Rule when needed. Inº y lim y→0* Vy
Evaluate the following limits lim x csc X |x→0
The function f has a root of multiplicity 2 at r if f(r) = f'(r) = 0 and f"(r) ≠ 0. In this case, a slight modification of Newton’s method, known as the modified (or accelerated) Newton’s method, is given by the formulaa. Verify that 0 is a root of multiplicity 2 of the function f(x) = e2 sin
Explain in words and express mathematically the inverse relationship between differentiation and integration as given by Part 1 of the Fundamental Theorem of Calculus.
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) = e2x
Determine the following indefinite integrals. Check your work by differentiation.∫(csc 3φ cot 3φ dφ
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) = 2x5 - 5x4 - 10x3 + 4 on [-2, 4]
Consider the integrala. Evaluate the right Riemann sum for the integral with n = 3.b. Use summation notation to express the right Riemann sum in terms of a positive integer n.c. Evaluate the definite integral by taking the limit as n→∞ of the Riemann sum of part (b).d. Confirm the result of
Find ∫cos2 x dx.
Does a left Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and increasing on an interval [a, b]? Explain
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) = xe1-x/2 on [0, 5]
Use the derivative f' to determine the x-coordinates of the local maxima and minima of f , and the intervals of increase and decrease. Sketch a possible graph of (f is not unique).f'(x) = (x - 1)(x + 2)(x + 4)
A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at an angle θ (see figure). What angle θ maximizes the cross sectional area of the gutter? 3 in 3 in ө 3 in Cross-sectional area
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 2r – 1 + 2x lim x2 х—0
Evaluate the following limits. 2 tan x lim xT/2 sec?x
To approximate the reciprocal of a number a without using division, we can apply Newton’s method to the function f(x) = 1/x - a.a. Verify that Newton’s method gives the formula xn + 1 = (2 - axn)xn.b. Apply Newton’s method with a = 7 using a starting value of your choice. Compute an
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) = 1/x3
Determine the following indefinite integrals. Check your work by differentiation.∫(3t2 + sec2 2t) 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) = -x3 + 9x on [-4, 3]
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)x on [0.1, 1]
An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions.
Determine whether the following statements are true and give an explanation or counterexample.a. If the zeros of f' are -3, 1, and 4, then the local extrema of f are located at these points.b. If the zeros of f" are -2 and 4, then the inflection points of f are located at these points.c. If the
A crank of radius r rotates with an angular frequency v. It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the functionFor fixed v and r, find the values of θ, with 0 ≤ θ ≤ 2π, for which
Suppose the functions f and g are continuous on [a, b] and differentiable on (a, b), where g(a) ≠ g(b). Then there is a point c in (a, b) at whichThis result is known as the Generalized (or Cauchy’s) Mean Value Theorem.a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to
Evaluate the following limits. Use l’Hôpital’s Rule when needed. lim 20 cot 30
Evaluate the following limits. х x² – In (2/x) lim Зx2 + 2х х—о
Let a > 0 be given and suppose we want to approximate √a using Newton’s method. a. Explain why the square root problem is equivalent to finding the positive root of f(x) = x2 - a.b. Show that Newton’s method applied to this function takes the form (sometimes called the Babylonian
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) = sin2 x
Determine the following indefinite integrals. Check your work by differentiation. sin 0 – 1 S1 cos e CoS
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) = 2x3 + 3x2 - 12x + 1 on [-2, 4]
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) = x2/3 on [-8, 8]
Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. Зх — 5 f(x) x? – 1
Suppose that a blood test for a disease is given to a population of N people, where N is large. At most, N individual blood tests must be done. The following strategy reduces the number of tests. Suppose 100 people are selected from the population and their blood samples are pooled. One test
Suppose f'(x) < 0 < f"(x), for x < a, and f'(x) > 0 > f"(x), for x > a. Prove that f is not differentiable at a. Assume f is differentiable at a and apply the Mean Value Theorem to f'.) More generally, show that if f' and f" change sign at the same point, then f is not
Evaluate the following limits. Use l’Hôpital’s Rule when needed. lim (Vx? + x + 1 – Vx² – x) )
Evaluate the following limits. In (3x + 5e*) lim x→0 In (7x + 3e2*)
Verify by graphing that the graphs of y = ex and y = x have no points of intersection, whereas the graphs of y = ex/3 and y = x have two points of intersection. Approximate the value of a > 0 such that the graphs of y = ex/a and y = x have exactly one point of intersection.
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) = 2x + 1
Determine the following indefinite integrals. Check your work by differentiation.∫(sec2 θ + sec θ tan θ) dθ
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) = x√4 - x2 on [-2, 2]
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) = sin 3x on [-π/4, π/3]
Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. Зх — 5 f(x) x2 - 1
Among all right circular cones with a slant height of 3, what are the dimensions (radius and height) that maximize the volume of the cone? The slant height of a cone is the distance from the outer edge of the base to the vertex.
The Jamaican sprinter Usain Bolt set a world record of 9.58 s in the 100-m dash in the summer of 2009. Did his speed ever exceed 37 km/hr during the race? Explain.
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 3 sin? 20 lim 02
Evaluate the following limits. In (3x + 5) lim x→o In (7x + 3) + 1
Verify by graphing that the graphs of y = sin x and y = x/2 have one point of intersection, for x > 0, whereas the graphs of y = sin x and y = x/9 have three points of intersection, for x > 0. Approximate the value of a such that the graphs of y = sin x and y = x/a have exactly two points of
Approximate the change in the magnitude of the electrostatic force between two charges when the distance between them increases from r = 20 m to r = 21 m (F(r) = 0.01/r2).
Determine the following indefinite integrals. Check your work by differentiation.∫(2 sec2 2v dv
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) = -x2 - x + 2 on [-4, 4]
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) = x/(x2 + 3)2 on [-2, 2]
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