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mathematics
calculus early transcendentals

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

Evaluate the following limits or state that they do not exist. cos x – lim 1 sin x х—0
Evaluate the following limits or state that they do not exist. 1 2 + sin 0 lim sin 0
Evaluate the following limits or state that they do not exist. sin x – 1 lim x→n/2 Vsin x – 1
Evaluate the following limits or state that they do not exist. sin? x + 6 sin x + 5 lim х>3п/2 sin? x – 1
Evaluate the following limits or state that they do not exist. cos x + 3 cos x + 2 lim cos x + 1 х—п
Use the continuity of the absolute value function to determine the interval(s) on which the following functions are continuous.h(x) = |x2 + 2x + 5|
Use the continuity of the absolute value function to determine the interval(s) on which the following functions are continuous. h(x) = Vx – 4
Use the continuity of the absolute value function to determine the interval(s) on which the following functions are continuous. x + 4 g(x) x² – 4 ||
Use the continuity of the absolute value function to determine the interval(s) on which the following functions are continuous.f(x) = |x2 + 3x - 18|
Prove that the absolute value function |x| is continuous for all values of x. (Hint: Using the definition of the absolute value function, compute lim x and lim x.) |х—0- х>0*
Determine whether the following statements are true and give an explanation or counterexample.a. If a function is left-continuous and right-continuous at a, then it is continuous at a.b. If a function is continuous at a, then it is left-continuous and right-continuous at a.c. If a < b and f(a)
a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.b. Use a graphing utility to find all the solutions to the equation on the given interval.c. Illustrate your answers with an appropriate graph.x ln x - 1 = 0; (1, e)
a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.b. Use a graphing utility to find all the solutions to the equation on the given interval.c. Illustrate your answers with an appropriate graph.x + ex = 0; (-1, 0)
a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.b. Use a graphing utility to find all the solutions to the equation on the given interval.c. Illustrate your answers with an appropriate graph.-x5 - 4x2 + 2√x + 5 = 0; (0, 3)
a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.b. Use a graphing utility to find all the solutions to the equation on the given interval.c. Illustrate your answers with an appropriate graph.x3 - 5x2 + 2x = -1; (-1, 5)
a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.b. Use a graphing utility to find all the solutions to the equation on the given interval.c. Illustrate your answers with an appropriate graph.√x4 + 25x3 + 10 = 5; (0, 1)
a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.b. Use a graphing utility to find all the solutions to the equation on the given interval.c. Illustrate your answers with an appropriate graph.2x3 + x - 2 = 0; (-1, 1)
You are shopping for a $150,000, 30-year (360-month) loan to buy a house. The monthly payment iswhere r is the annual interest rate. Suppose banks are currently offering interest rates between 6% and 8%.a. Use the Intermediate Value Theorem to show there is a value of r in (0.06, 0.08)—an
Suppose $5000 is invested in a savings account for 10 years (120 months), with an annual interest rate of r, compounded monthly. The amount of money in the account after 10 years is A(r) = 5000(1 + r/12)120.a. Use the Intermediate Value Theorem to show there is a value of r in (0, 0.08)—an
Determine the interval(s) on which the following functions are continuous; then analyze the given limits. f(x) : lim f(x) x→0' et
Determine the interval(s) on which the following functions are continuous; then analyze the given limits. f(x) et lim f(x); limf(x) х>0- 1 - ets х>0+*
Determine the interval(s) on which the following functions are continuous; then analyze the given limits. In x ; lim_f(x) f(x) х х>1 sin 1
Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 1 + sin x f(x) lim_ f(x); lim f(x) х—п/2 х—4п/3 cos x
Determine the interval(s) on which the following functions are continuous; then analyze the given limits.
Determine the interval(s) on which the following functions are continuous; then analyze the given limits. f(x) = csc x; lim f(x); lim_f(x) х>п/4' х>2т
Evaluate each limit and justify your answer. 12 + 5 lim -2 1 + Vr² + 5
Evaluate each limit and justify your answer. lim Vx? + 7 .2
Evaluate each limit and justify your answer. lim (x² – 4 + Vx² – 9) x→-1
Evaluate each limit and justify your answer. 4х + 10 lim V 2х — 2 х—2
Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left continuity at the endpoints.f(z) = (z - 1)3/4
Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left continuity at the endpoints.f(x) = (2x - 3)2/3
Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left continuity at the endpoints.f (t) = (t2 - 1)3/2
Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left continuity at the endpoints.f (x) = 3√x2 - 2x - 3
Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left continuity at the endpoints.g(x) = √x4 - 1
Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left continuity at the endpoints.f(x) = √2x2 - 16
Leta. Use the continuity checklist to show that f is not continuous at 0.b. Is f continuous from the left or right at 0?c. State the interval(s) of continuity. х3 + 4х + 1 if x < 0 F(х) (2r³ 2к3 if x > 0.
Leta. Use the continuity checklist to show that f is not continuous at 0.b. Is f continuous from the left or right at 0?c. State the interval(s) of continuity. (x* + 4x + 1 if x < 0 .3 f(x) 2r3 if x > 0.
Determine the intervals of continuity for the following functions.The graph of Exercise 12 УА y, 5 - y = f(x) 4 3 2 2 3 4 5 x
Determine the intervals of continuity for the following functions.The graph of Exercise 11 5 y = f(x) 4 3 2 3 4 5
Determine the intervals of continuity for the following functions.The graph of Exercise 10 5 4 y = f(x) 3 3 4 5
Determine the intervals of continuity for the following functions.The graph of Exercise 9 y, 5 4 y = f(x) 3 2 1 х 3 4 5
Evaluate each limit and justify your answer. 1/3 lim х>0 V16x + 1 – 1
Evaluate each limit and justify your answer. 2 sin x lim In
Evaluate each limit and justify your answer. lim tan Vi - 2
Evaluate each limit and justify your answer. .3 2x2 8x lim x→4
Evaluate each limit and justify your answer. 3 2х + 1 lim х># х
Evaluate each limit and justify your answer. lim x→1
Evaluate each limit and justify your answer. 3 lim 2x - 4x2 – 50 x→2
Evaluate each limit and justify your answer.(x8 - 3x6 - 1)40 lim X*
Use Theorem 2.10 to determine the intervals on which the following functions are continuous. t + 2 f(t) .2 4
Use Theorem 2.10 to determine the intervals on which the following functions are continuous. f(x) .2 – 4
Use Theorem 2.10 to determine the intervals on which the following functions are continuous. x² – 4x + 3 s(x) x² – 1
Use Theorem 2.10 to determine the intervals on which the following functions are continuous. x' + 6x + 17 x² – 9 f(x)
Use Theorem 2.10 to determine the intervals on which the following functions are continuous. 3x2 – 6x + 7 Зх2 8(x) x² + x + 1 .2
Use Theorem 2.10 to determine the intervals on which the following functions are continuous.p(x) = 4x5 - 3x2 + 1
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. .2 if x + -1 ; a = f(x) -1 х+ 1 if x = -1
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 5х — 2 4 f(x) x² – 9x + 20
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. x² - 4x + 3 if x + 3 f(x) = х- 3 ; a = 3 if x =
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. .2 - 1 if x + 1 f(x) ; a = 1 3 if x = 1
Use analytical methods to identify all the asymptotes of Then confirm your results by locating the asymptotes with a graphing utility. In(9 — х) Г) 2еx - е х
Evaluate and Then state the horizontal asymptote(s) of f. Confirm your findings by plotting f. lim f(x)
Evaluate and Then state the horizontal asymptote(s) of f. Confirm your findings by plotting f. lim f(x)
Use the following instructions to determine the end behavior of a. Evaluate by first dividing the numerator and denominator by e3x.b. Evaluate by first dividing the numerator and denominator by e2x.c. Give the horizontal asymptote(s).d. Graph f to confirm your work in parts (a)–(c). e* + e2x ,2х
a. Is it possible for a rational function to have both slant and horizontal asymptotes? Explain.b. Is it possible for an algebraic function to have two distinct slant asymptotes? Explain or give an example.
Suppose f (x) = p(x)/q(x) is a rational function, wherea. Prove that if m = n, then b. Prove that if m 6 n, then p(x) = amr™ q(x) = b„x" + b,- 1x"- + · · · + bzx² + bjx + bo, am + 0, and b, + 0. .т— 1 + · · · + a2x² + a ¡x + ao, аmх" + аm- |X" п
A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2, 4, 6, 8, . . . .} is specified by the function f (n) = 2n, where n = 1, 2, 3, . . . . . The limit of such a sequence is provided the limit exists. All the limit laws for limits at
A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2, 4, 6, 8, . . . .} is specified by the function f (n) = 2n, where n = 1, 2, 3, . . . . . The limit of such a sequence is provided the limit exists. All the limit laws for limits at
A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2, 4, 6, 8, . . . .} is specified by the function f (n) = 2n, where n = 1, 2, 3, . . . . . The limit of such a sequence is provided the limit exists. All the limit laws for limits at
A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2, 4, 6, 8, . . . .} is specified by the function f (n) = 2n, where n = 1, 2, 3, . . . . . The limit of such a sequence is provided the limit exists. All the limit laws for limits at
If a function f represents a system that varies in time, the existence of means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.The amplitude of an oscillator is given by a(t) = 2(t + sin
If a function f represents a system that varies in time, the existence of means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.The population of a colony of squirrels is given by p(t) =
If a function f represents a system that varies in time, the existence of means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.The value of an investment in dollars is given by v(t) =
If a function f represents a system that varies in time, the existence of means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.The amount of drug (in milligrams) in the blood after an IV
If a function f represents a system that varies in time, the existence of means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.The population of a culture of tumor cells is given by p(t) =
If a function f represents a system that varies in time, the existence of means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.The population of a bacteria culture is given by p(t) = 2500/t
Find the vertical and horizontal asymptotes of cos x + 2Vx f(x) VE
Find the vertical and horizontal asymptotes of f(x) = e1/x.
Sketch a possible graph of a function f that satisfies all the given conditions. Be sure to identify all vertical and horizontal asymptotes. lim f(x) = - 00, lim f(x) = 1, = 0, lim f(x) lim f(x) = -2
Sketch a possible graph of a function f that satisfies all the given conditions. Be sure to identify all vertical and horizontal asymptotes.f(-1) = -2, f(1) = 2, f(0) = 0, = 1, | l lim f(x) lim f(x) = -1 -1 x -00
The hyperbolic sine function is defined as sinh x = ex - e-x/2a. Determine its end behavior by analyzing and b. Evaluate sinh 0. Use symmetry and part (a) to sketch a plausible graph for y = sinh x. lim sinh x lim sinh x. X -00
The hyperbolic cosine function, denoted cosh x, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh x = ex + e-x/2.a. Determine its end behavior by analyzing and b. Evaluate cosh 0. Use symmetry and part (a) to sketch a plausible graph for y =
Consider the graph y = sec-1 x (see Section 1.4) and evaluate the following limits using the graph. Assume the domain is {x: |x| ≥ 1}.a. b. -1 х lim sec х- lim secx х- ,-1 х>—оо
The central branch of f(x) = tan x is shown in the figure.a. Analyze tan x. Are these infinite limits or limits at infinity?b. Sketch a graph of g(x) = tan-1 x by reflecting the graph of f over the line y = x and use it to evaluate and lim tan x and x→T/2¯ lim x→-T/2* lim tan'x
a. Analyze and then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze f(x) = √|x| - √|x - 1| lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyze and then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyze and then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyze and then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyzeand then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x),
a. Analyzeand then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze f(x) = 16x2(4x2 - √16x4 + 1) lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyzeand then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyzeand then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyzeand then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
a. Analyzeand then identify any horizontal asymptotes.b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim f(x) and 1lim f(x), lim f(x) and lim f(x). хэа х—а +'
Determine whether the following statements are true and give an explanation or counterexample.a. The graph of a function never crosses one of its horizontal asymptotes.b. A rational function f has both (where L is finite) and c. The graph of a function can have at most two horizontal
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.f(x) = 50/e2x
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.f(x) = sin x
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.f(x) = |ln x|
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.f(x) = 1 - ln x
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.f(x) = 2x
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.f(x) = -3e-x

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