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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.f(x) = x3 - 33x2 + 216x - 2
What are the radius and area of the circle of maximum area that can be inscribed in an isosceles triangle whose two equal sides have length 1?
Verify that the functions f(x) = sin2 x and g(x) = -cos2 x have the same derivative. What can you say about the difference f - g? Explain.
Prove that for a ≠ 0. х ей lim х
Verify the following indefinite integrals by differentiation. ² cos x* dx = sin x + x² cos x³ dx 3 .3
Show that xx grows faster than bx as x→∞, for b > 1.
Verify the following indefinite integrals by differentiation. ·dx 3= (x² – 1)? 2(x² – 1)
a. For what values of b > 0 does bx grow faster than ex as x→∞?b. Compare the growth rates of ex and eax as x→∞, for a > 0.
What does net area measure?
Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2, and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.
Suppose A is an area function of f. What is the relationship between f and A?
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 5x2 + 2x lim x 00 – 5 Vx* – 1
Evaluate the following limits. ,Зх езx lim х—о Зезх + 5 ,Зх
Approximate the root of f(x) = x10 at x = 0 using Newton’s method with an initial approximation of x0 = 0.5. Make a table showing the first 10 approximations, the error in these approximations (which is |xn - 0| = |xn|), and the residual of these approximations (which is f(xn)). Comment on the
Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone with fixed height h = 6 m when its radius decreases from r = 10 m or = 9.9 m (S = pr√r2 + h2).
On which derivative rule is the Substitution Rule based?
Determine the following indefinite integrals. Check your work by differentiation.∫(sec2 (x - 1) 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 + 3 on [-3, 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) = cos2 x on [0, π]
Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.f(x) = 3x4 + 4x3 - 12x2
a. An isosceles triangle has a base of length 4 and two sides of length 212. Let P be a point on the perpendicular bisector of the base. Find the location P that minimizes the sum of the distances between P and the three vertices. b. Assume in part (a) that the height of the isosceles triangle
What is the geometric meaning of a definite integral if the integrand changes sign on the interval of integration?
Given the graph of the positive velocity of an object moving along a line, what is the geometrical representation of its displacement over a time interval [a, b]?
Suppose F is an antiderivative of f and A is an area function of f. What is the relationship between F and A?
Verify that the functions f(x) = tan2 x and g(x) = sec2 x have the same derivative. What can you say about the difference f - g? Explain.
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 1 - cos 6t lim 2t
Evaluate the following limits. tan x lim x→n /2 3/(2x T
Evaluate the following limits. 4x lim 3 – 2x2 + 6 TTX + 4 Tx3
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.f(x) = x2(x - 100) + 1
An object travels on the x-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4. a. How far does the object travel, for 0 ≤ t ≤ 4?b. What is the average value of v on the interval [0, 4]?c. True or false: The object would travel as far as in part (a) if it traveled at its average
Approximate the change in the volume of a right circular cone of fixed height h = 4 m when its radius increases from r = 3 m to r = 3.05 m (V(r) = πr2 h/3).
Determine the following indefinite integrals. Check your work by differentiation.∫(sin 4t - sin t/4) dt
Find the intervals on which f is increasing and decreasing. - 1 tan f(x) + 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) = (x + 1)4/3 on [-9, 7]
Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. F(x) - x + 1 15 .3
Why is the Substitution Rule referred to as a change of variables?
Under what conditions does the net area of a region equal the area of a region? When does the net area of a region differ from the area of a region?
Solve the problem in Exercise 37, but this time minimize the cost with respect to the smaller angle u between the underwater cable and the shore. (You should get the same answer.)Data from Exercise 37An island is 3.5 mi from the nearest point on a straight shoreline; that point is 8 mi from a power
Suppose you want to approximate the area of the region bounded by the graph of f(x) = cos x and the x-axis between x = 0 and x = π/2. Explain a possible strategy.
Explain in words and write mathematically how the Fundamental Theorem of Calculus is used to evaluate definite integrals.
Use geometry to evaluate the following definite integrals, where the graph of f is given in the figure.a.b.c.d. f(x) dx 0. f(x) dx 6.
a. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = x2 on [a, b] is the arithmetic mean of a and b; that is, c = (a + b)/2.b. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a, b], where 0 < a < b, is the geometric mean of a
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 13 – t? – 21 lim 1? – 4 →2
Evaluate the following limits. 3x4 — х2 Зх4 lim х—о 6х* + 12
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.f(x) = ln x - x2 + 3x - 1
Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 cm to h = 11.9 cm (V(h) = πr2 h).
The composite function f(g)x)) consists of an inner function g and an outer function f. If an integrand includes f(g)x)), which function is often a likely choice for a new variable u?
Determine the following indefinite integrals. Check your work by differentiation.∫(sin 2y + cos 3y) dy
Suppose that f(x) < 0 on the interval [a, b]. Using Riemann sums, explain why the definite integralis negative. Saf(x) dx
Find the intervals on which f is increasing and decreasing.f(x) = xe-x2/2
Explain how Riemann sum approximations to the area of a region under a curve change as the number of subintervals increases.
Let f(x) = c, where c is a positive constant. Explain why an area function of f is an increasing function.
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 - 10 on [-2, 3]
Use geometry to find the displacement of an object moving along a line for the time intervals (i) 0 ≤ t ≤ 5.(ii) 3 ≤ t ≤ 7.(iii) 0 ≤ t ≤ 8, where the graph of its velocity v = g(t) is given in the figure. 5 4 = g(t) 3 3 5 6 4-
Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. — —х3 — 2х? - 5х + 2 f(x) 2r2 3
An island is 3.5 mi from the nearest point on a straight shoreline; that point is 8 mi from a power station (see figure). A utility company plans to lay electrical cable underwater from the island to the shore and then underground along the shore to the power station. Assume that it costs $2400/mi
Find a suitable substitution for evaluating and explain your choice. |S tan x sec? x dx
Use graphs to evaluate r2 " sin x dx 277 J" cos x dx. 277
Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length Δx? List the grid points x0, x1, x2, x3, and x4. Which points are used for the left, right, and midpoint Riemann sums?
Consider the quadratic function f(x) = Ax2 + Bx + C, where A, B, and C are real numbers with A ≠ 0. Show that when the Mean Value Theorem is applied to f on the interval [a, b], the number c guaranteed by the theorem is the midpoint of the interval.
The linear function f(x) = 3 - x is decreasing on the interval [0, 3]. Is the area function for f (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Use Newton’s method to approximate the x-coordinate of the inflection points of f(x) = 2x5 - 6x3 - 4x + 2 to six digits.
Use geometry to evaluate Complete the square. SV8r – x² dx. '8х — х2 dx.
When using a change of variables u = g(x) to evaluate the definite integral how are the limits of integration transformed? | SF(8(x))g'(x) dx,
Explain how the notation for Riemann sums, corresponds to the notation for the definite integral, η Σ) Δ Xr, k=1 Sif(x) dx.
Suppose the interval [2, 6] is partitioned into n = 4 subintervals with grid points x0 = 2, x1 = 3, x2 = 4, x3 = 5, and x4 = 6. Write, but do not evaluate, the left, right, and midpoint Riemann sums for f(x) = x2.
Evaluateand ГЗх2 dx 3л Зx2 dx. 2 3. -2
The manager of a bagel bakery collects the following production rate data (in bagels per minute) at seven different times during the morning. Estimate the total number of bagels produced between 6:00 and 7:30 a.m., using a left and right Riemann sum. Production rate Time of day (A.M.) (bagels/min)
Evaluate the following limits. (3x + 2 – 2 lim x→2
If the change of variables u = x2 - 4 is used to evaluate the definite integralwhat are the new limits of integration? Sf(x) dx,
Give a geometrical explanation of why Saf(x) dx = 0.
Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval [a, b]? Explain.
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. f(x) 5 4 20
Approximate the change in the atmospheric pressure when the altitude increases from z = 2 km to z = 2.01 km (P(z) = 1000 e-z/10).
Determine the following indefinite integrals. Check your work by differentiation. ` 12r8 – t dt ,3
Find the intervals on which f is increasing and decreasing. 8x3 15x? f(x) 3 4-
a. Find the critical points of the following functions on the domain or on the given interval.b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither.f(x) = (sin-1 x)(cos-1 x) on [0, 1]
The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.A: v(t) = sin t, s(0) = 0; B: V(t) = cos t, S(0) = 0
Graph the function f(x) = 60x5 - 901x3 + 27x in the window [-4, 4] × [-10000, 10000]. How many extreme values do you see? Locate all the extreme values by analyzing f'.
Let n be a positive integer. Use graphical and/or analytical methods to verify the following limits.a.
Use analytical methods to evaluate the following limits. lim (log2x – log3 x)
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.a(t) = 2e-t/6; v(0) = 1, s(0) = 0
Sketch the graph of a function that is continuous on (-∞, ∞) and satisfies the following sets of conditions.f"(x) > 0 on (-∞, -2); f"(x) < 0 on (-2, 1); f"(x) > 0 on (1, 3); f"(x) < 0 on (3, ∞)
The functions f(x) = (xx)x and g(x) = x(xx)are different functions. For example, f(3) = 19,683 and g(3) ≈ 7.6 × 1012. Determine whetherandare indeterminate forms and evaluate the limits. lim f(x) lim g(x)
Use analytical methods to evaluate the following limits. log2 x lim xo log3 x
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.a(t) = 3 sin 2t; v(0) = 1, s(0) = 10
Sketch the graph of a function that is continuous on (-∞, ∞) and satisfies the following sets of conditions.f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f "(0) = 0
Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work.g(x) = 1/(e-x - 1)
A rectangular sheet of paper of width a and length b, where 0 < a < b, is folded by taking one corner of the sheet and placing it at a point P on the opposite long side of the sheet (see figure). The fold is then flattened to form a crease across the sheet. Assuming that the fold is made so
Interpret the Mean Value Theorem when it is applied to any linear function.
Use Newton’s method to approximate the roots of f(x) = e-2x + 2ex - 6 to six digits. Make a table showing the first five approximations for each root using an initial estimate of your choice.
Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule. x4 – Vx lim x→0 2x + x 4
Use analytical methods to evaluate the following limits. lim x/In x x>0+
Evaluate the following limits. x? – 4x + 4 lim sin п. sin² (Tx)
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.a(t) = 2 cos t; v(0) = 1, s(0) = 0
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. f(x) = e*
Sketch the graph of a function that is continuous on (-∞, ∞) and satisfies the following sets of conditions.f(-2) = f"(-1) = 0; f'(-3/2) = 0; f(0) = f'(0) = 0; f(1) = f'(1) = 0
Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule. 2x - x + 1 2r5 lim х- 5. x 00 5x° + x
Approximate the change in the volume of a sphere when its radius changes from r = 5 ft to r = 5.1 ft or r = 5.1 ft (V(r) = 4/3 πr3).
Use analytical methods to evaluate the following limits. 1 lim Vx – 1 x→1*
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