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mathematics
calculus early transcendentals

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

Assume that postage for sending a first-class letter in the United States is $0.44 for the first ounce (up to and including 1 oz) plus $0.17 for each additional ounce (up to and including each additional ounce).a. Graph the function p = f(w) that gives the postage p for sending a letter that weighs
Use the zoom and trace features of a graphing utility to approximate the following limits. 6* – 3* lim- 0 хIn 2 х
Use the zoom and trace features of a graphing utility to approximate the following limits. 9(V2r – x* – V3) 1 - x3/4 t/g* lim
Use the zoom and trace features of a graphing utility to approximate the following limits. 18(Vx – 1) lim x³ - 1
Use the zoom and trace features of a graphing utility to approximate the following limits.limx→0 x sin1/x
For any real number x, the ceiling function [x] is the smallest integer greater than or equal to x.a. Graph the ceiling function y = [x], for -2 ≤ x ≤ 3.b. Evaluate limx→2- [x], limx→1+ [x], and limx→1.5 [x].c. For what values of a does limx→a [x] exist? Explain.
For any real number x, the floor function (or greatest integer function) [x] is the greatest integer less than or equal to x (see figure).a. Compute limx→-1- [x], limx→-1+ [x], limx→2- [x], and limx→2+ [x].b. Compute limx→2.3- [x], limx→2.3+ [x], and limx→2.3 [x].c. For a given
Let f(x) = |x|/x, for x ≠ 0.a. Sketch a graph of f on the interval [-2, 2].b. Does limx→0 f(x) exist? Explain your reasoning after first examining limx→0- f(x) and limx→0+ f(x).
Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001.limh→0 ln (1 + h)/h
Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001.limh→0 2h - 1/h
Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001.limh→0 (1 + 3h)2/h
Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001.limh→0(1 + 2h)1/h
Sketch the graph of a function with the given properties. You do not need to find a formula for the function. p(0) = 2, lim p(x) = 0, lim p(x) does not exist, P(2) = lim p(x) = 1 %3D X-2+
Sketch the graph of a function with the given properties. You do not need to find a formula for the function. h(-1) = 2, lim_ h(x) = 0, lim h(x) = 3, h(1) = lim h(x) = 1, lim h(x) = 4
Sketch the graph of a function with the given properties. You do not need to find a formula for the function. g(1) = 0, g(2) = 1, g(3) = -2, lim g(x) = 0, %3D %3D lim g(x) = -1, lim g(x) I-3* = -2
Sketch the graph of a function with the given properties. You do not need to find a formula for the function. f(1) = 0.f(2) = 4.f(3) = 6, lim_f(x) = -3, lim f(x) = 5
Determine whether the following statements are true and give an explanation or counterexample.a. The value of limx→3 x2 - 9/x - 3 does not exist.b. The value of limx→a f(x) is always found by computing f(a).c. The value of limx→a f(x) does not exist if f(a) is undefined.d. limx→0 √x =
a. Create a table of values of tan (3/x) for x = 12/π, 12/13π2,12/15π2, ... , 12/(11π). Describe the general pattern in the values you observe.b. Use a graphing utility to graph y = tan (3/x). Why do graphing utilities have difficulty plotting the graph near x = 0?c. What do you conclude about
a. Create a table of values of sin (1/x), for x = 2/π, 2/3π, 2/5π, 2/7π, 2/9π, and 2/11π. Describe the pattern of values you observe.b. Why does a graphing utility have difficulty plotting the graph of y = sin (1/x) near x = 0 (see figure)?c. What do you conclude about limx→0sin (1/x)? х -1
Use the graph of g in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.a. g(-1)b. limx→-1- g(x)c. limx→-1+ g(x)d. limx→-1 g(x)e. g(1)f. limx→1 g(x)g. limx→3 g(x)h. g(5)i. limx→5- g(x) УА 5- у3 в) 3 4 5 х -3 -2 -
Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.a. f(1)b. limx→1- f(x)c. limx→1+ f(x)d. limx→1 f(x)e. f(3)f. limx→3- f(x)g. limx→3+ f(x)h. limx→3 f(x)i. f(2)j. limx→2- f(x)k. limx→2+ f(x)l.
Use the graph of g in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.a. g(2)b. limx→2- g(x)c. limx→2+ g(x)d. limx→2 g(x)e. g(3)f. limx→3- g(x)g. limx→3+ g(x)h. g(4)i. limx→4 g(x) y = g(x) 3+ 3
Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.a. f(1)b. limx→1- f(x)c. limx→1+ f(x)d. limx→1 f(x) УА у 3Га) х
Let g(x) = x - 100/√x - 10. Use tables and graphs to make a conjecture about the values of limx→100+ g(x), limx→100- g(x), and limx→100 g(x) or state that they do not exist.
Let f(x) = x2 - 25/x - 5. Use tables and graphs to make a conjecture about the values of limx→5+f(x), limx→5-f(x), and limx→5 f(x) or state that they do not exist.
Let g(x) = 3 sin x - 2 cos x + 2/x.a. Graph of g to estimate limx→0 g(x).b. Evaluate g(x) for values of x near 0 to support your conjecture in part (a).
Given the function f(x) = 1 - cos x and the points A(π/2, f(π/2)), B(π/2 + 0.05, f(π/2 + 0.05)), C(π/2 + 0.5, f(π/2 + 0.5)), and D(p, f(p)) (see figure), find the slopes of the secant lines through A and D, A and C, and A and B. Then use your calculations to make a conjecture about the slope
A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t) = 16t2. Assume the distance from the top of the cliff to the ground is 96 ft.a. When will the rock strike the ground?b. Make a table of average velocities and approximate the
A projectile is fired vertically upward and has a position given by s(t) = -16t2 + 128t + 192, for 0 ≤ t ≤ 9.a. Graph the position function, for 0 ≤ t ≤ 9.b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this
a. Graph the function f(x) = 4 - x2.b. Identify the point (a, f(a)) at which the function has a tangent line with zero slope.c. Consider the point (a, f(a)) found in part (b). Is it true that the secant line between (a - h, f(a - h)) and (a + h, f(a + h)) has slope zero for any value of h ≠ 0?
a. Graph the function f(x) = x2 - 4x + 3.b. Identify the point (a, f(a)) at which the function has a tangent line with zero slope.c. Confirm your answer to part (b) by making a table of slopes of secant lines to approximate the slope of the tangent line at this point.
For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.f(x) = x3 - x at x = 1
For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.f(x) = ex at x = 0
For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.f(x) = 3 cos x at x = π/2
For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.f(x) = 2x2 at x = 2
For the following position functions, make a table of average velocities similar and make a conjecture about the instantaneous velocity at the indicated time.s(t) = 20/(t + 1) at t = 0
For the following position functions, make a table of average velocities similar and make a conjecture about the instantaneous velocity at the indicated time.s(t) = 40 sin 2t at t = 0
For the following position functions, make a table of average velocities similar and make a conjecture about the instantaneous velocity at the indicated time.s(t) = 20 cos t at t = π/2
For the following position functions, make a table of average velocities similar and make a conjecture about the instantaneous velocity at the indicated time.s(t) = -16t2 + 80t + 60 at t = 3
Consider the position function s(t) = 3 sin t that describes a block bouncing vertically on a spring. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = π/2.Time interval
Consider the position function s(t) = -16t2 + 100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = 3.Time interval Average velocity[2,
The following table gives the position s(t) of an object moving along a line at time t. Determine the average velocities over the time intervals [2, 2.01], [2, 2.001], and [2, 2.0001]. Then make a conjecture about the value of the instantaneous velocity at t = 2. 2 2.0001 2.001 2.01 55.99959984
The following table gives the position s(t) of an object moving along a line at time t. Determine the average velocities over the time intervals [1, 1.01], [1, 1.001], and [1, 1.0001]. Then make a conjecture about the value of the instantaneous velocity at t = 1. 1.001 1.0001 1 1.01 64 s(t)
Determine the end behavior of the following functions. f(x) Vor? V9x² + x
Evaluate the following limit or state that it does not exist. 4x + 1 f(x) = .3 1 – x'
Evaluate the following limit or state that it does not exist. 1 lim r→o In r + 1
Evaluate the following limit or state that it does not exist. x + 2) lim (3 tan¬ -1
Evaluate the following limit or state that it does not exist. -2z lim Z.
Evaluate the following limit or state that it does not exist. lim (-3x³ + 5)
Evaluate the following limit or state that they do not exist. x* – 1 lim .5 x + 2
Evaluate the following limits or state that they do not exist. 2x – 3 lim x→0 4x + 10
Analyze the following limit. lim x→0¯ tan x
Analyze the following limits. и lim u→0+ sin u
Analyze the following limits. lim x-3- x – 3x .2
Analyze the following limits. х — 5 lim х>-5* х
Analyze the following limits. х lim 2 х-5 x(х — 5)
Assume the function g satisfies the inequality 1 ≤ g(x) ≤ sin2 x + 1, for x near 0. Use the Squeeze Theorem to find lim g(x). x→0
a. Use a graphing utility to illustrate the inequalitieson [-1, 1].b. Use part (a) and the Squeeze Theorem to explain why sin x cos x < cos x VI sin x lim 1. ||
Determine the following limits analytically. Vsin x lim x→T/2 X + T/2
Determine the following limits analytically. Vx - 3 lim х>81 х — 81
Determine the following limits analytically. sin?0 – cos?0 lim 0→T/4 sin 0 – cos 0
Determine the following limits analytically. p³ – 1 lim p→1 p – 1
Determine the following limits analytically. - 81 х lim х—3 х — 3
Determine the following limits analytically. x3 lim 7x2 + 12x x→4
Evaluate the following limits. 3 + 2x + 4xr? lim
Graph the function f(x) = 1/x2 - x using a graphing utility with the window [-1, 2] × [-10, 10]. Use your graph to determine the following limits.a. limx→0- f(x)b. limx→0+ f(x)c. limx→1- f(x)d. limx→1+ f(x)
Determine the following limits. 11 lim 3x' x→-0∞
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. g(x) a = 3 х
Consider the position function s(t) = -4.9t2 + 30t + 20 (Exercise 10). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = 2.
The function f in the figure satisfies For each value of ε, find all values of δ > 0 such thata. ε = 2b. ε = 1c. For any ε > 0, make a conjecture about the corresponding value of δ satisfying (3). = 5. lim f(x) |f(x) – 5| < ɛ whenever 0 < |x – 4| < 8. (3)
Let f(x) = 1 - cos(2x - 2)/(x - 1)2.a. Graph f to estimate limx→1 f(x).b. Evaluate f(x) for values of x near 1 to support your conjecture in part (a).
Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions. g(5) = -1, lim g(x) = lim g(x) |8(2) = 1, - 00 lim g(x) = 0, 00
Determine the following limits analytically. 1 – 1/3 lim -1/3 (3t – 1)² 2
Evaluate the following limits. Cos o lim
Determine the points at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. 5 y = f(x) 3 3 4 5
The function f in the figure satisfies Determine the largest value of δ > 0 satisfying each statement.a. If 0 < |x - 3| < δ, then |f(x) - 6| < 3.b. If 0 < |x - 3| < δ, then |f(x) - 6| < 1. lim f(x) = 6. x→3 УА y = f(x) 1 3 х
Let f(x) = x2 - 4/x - 2.a. Calculate f(x) for each value of x in the following table.b. Make a conjecture about the value of limx→2 x2 - 4/x - 2. 1.9 1.99 1.999 1.9999 x? - 4 S(x) 2.01 2.001 2.1 2.0001 x? - 4 S(x)
The graph of p in the figure has vertical asymptotes at x = -2 and x = 3. Analyze the following limits.a. limx→-2- p(x)b. limx→-2+ p(x)c. limx→-2 p(x)d. limx→3- p(x)e. limx→3+ p(x)f. limx→3 p(x) УА х -2 y = p(x)
Determine the points at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. y A y = f(x) 4 2 - 4 5 x 2 3 3.
The function f in the figure satisfies Determine the largest value of δ > 0 satisfying each statement.a. If 0 < |x - 4| < δ, then |f(x) - 5| < 1.b. If 0 < |x - 4| < δ, then |f(x) - 5| < 0.5. lim f(x) = 5. УА y. y = f(x) 5 8. х 8.
Let f(x) = x3 - 1/x - 1.a. Calculate f(x) for each value of x in the following table.b. Make a conjecture about the value of limx→1 x3 - 1/x - 1. 0.9 0.99 0.999 0.9999 r' - 1 f(x) %3D 1.01 1.001 1.1 1.0001 x' - 1 f(x)
Determine the following limits analytically. 1 — х2 lim х—1x2 — 8х +7
Evaluate the following limits. cos x lim Vx
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 2x2 + 3x + 1 f(x) ; a = 5 х2 + 5х
Consider the position function s(t) = -16t2 + 100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.
Let f(x) = x3 + 3 and note that For each value of ε, use a graphing utility to find all values of δ > 0 such that |f(x) - 3| < ε whenever 0 < |x - 0| < δ. Sketch graphs illustrating your work.a. ε = 1b. ε = 0.5 lim f(x) = 3. x-
Let g(t) = t - 9/√t - 3.a. Make two tables, one showing values of g for t = 8.9, 8.99, and 8.999 and one showing values of g for t = 9.1, 9.01, and 9.001.b. Make a conjecture about the value of limt→9 t - 9/√t - 3.
Graph the function f(x) = e-x/x(x + 2)2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.a. limx→-2+ f(x)b. limx→-2 f(x)c. limx→0- f(x)d. limx→0+ f(x)
Determine the following limits analytically. V3x + 16 – 5 lim х — 3
Evaluate the following limits. 4 sint x3 100 lim ( 5 + x² x→-∞
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 2х2 + 3x + 1 f(x) ; a = -5 х? + 5х
Let g(x) = 2x3 - 12x2 + 26x + 4 and note that For each value of ε, use a graphing utility to find all values of δ > 0 such that |g(x) - 24| < ε whenever 0 < |x - 2| < δ. Sketch graphs illustrating your work.a. ε = 1b. ε = 0.5 24. lim g(x)
Let f(x) = (1 + x)1/x.a. Make two tables, one showing values of f for x = 0.01,0.001, 0.0001, and 0.00001 and one showing values of f for x = -0.01, -0.001, -0.0001, and -0.00001. Round your answers to five digits.b. Estimate the value of limx→0 (1 + x)1/x.c. What mathematical constant does
Consider the position function s(t) = sin pt representing the position of an object moving along a line on the end of a spring. Sketch a graph of s together with a secant line passing through (0, s(0)) and (0.5, s(0.5)). Determine the slope of the secant line and explain its relationship to the
Consider the position function s(t) = -16t2 + 128t (Exercise 9). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = 1. Time interval [1,2] [1, 1.5] [1, 1.1] [1, 1.01]| [1, 1.001] Average velocity
Sketch a possible graph of a function f, together with vertical asymptotes, satisfying all the following conditions on [0, 4]. f(1) = 0, lim f(x) f(3) is undefined, lim f(x) = 1, lim f(x) lim f(x) : = - ∞, 00 = 0,
Determine the following limits analytically. lim 2 Vx + 1 х—3 х — 3
Determine the following limits. lim x12
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.f(x) = √x - 2; a = 1
The function f in the figure satisfies For each value of ε, find all values of δ > 0 such that|f(x) - 3| < ε whenever 0 < |x - 2| < δ. (2)a. ε = 1b. ε = 1/2c. For any ε > 0, make a conjecture about the corresponding value of δ

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