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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Let f(x) = x - 2/ln|x - 2|.a. Graph f to estimate limx→2 f(x).b. Evaluate f(x) for values of x near 2 to support your conjecture in part (a).
Let g(x) = e2x - 2x - 1/x2.a. Graph g to estimate limx→0 g(x).b. Evaluate g(x) for values of x near 0 to support your conjecture in part (a).
Determine the following limits analytically. 7x² + 12x lim
The graph of h in the figure has vertical asymptotes at x = -2 and x = 3. Analyze the following limits.a. limx→-2- h(x)b. limx→-2+ h(x)c. limx→-2 h(x)d. limx→3- h(x)e. limx→3+ h(x)f. limx→3 h(x) УА -2 х 3 y = h(x)
Use the graph of f in the figure to find the following values or state that they do not exist.a. f(2)b. limx→2f(x)c. limx→4f(x)d. limx→5f(x) y = f(x) 4 5 6 3
The function f in the figure satisfies Determine the largest value of δ > 0 satisfying each statement.a. If 0 < |x - 2| < δ, then |f(x) - 4| < 1.b. If 0 < |x - 2| < δ, then |f(x) - 4| < 1/2. lim f(x) = 4. |х—2 УА y = f(x) 4 1 + 1 4
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. УА 5 4 y = f(x) 2 1 + х 3 4 3.
Evaluate the following limits. 1 lim ( 5 + 10 2. х* х
Determine the following limits analytically.where x is constant. – V5x V5x + 5h lim h→0
The graph of g in the figure has vertical asymptotes at x = 2 and x = 4. Analyze the following limits.a. limx→2- g(x)b. limx→2+ g(x)c. limx→2 g(x)d. limx→4- g(x)e. limx→4+ g(x)f. limx→4 g(x) y, х у 3 g(x) y =
Use the graph of f in the figure to find the following values or state that they do not exist.a. f(1)b. limx→1f(x)c. f(0)d. limx→0f(x) 4 y = f(x) -2
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. УА 5 4 y = f(x) |3D 3 2 х 2 3 4 5 1
The function f in the figure satisfiesDetermine the largest value of δ > 0 satisfying each statement.a. If 0 < |x - 2| < δ, then |f(x) - 5| < 2.b. If 0 < |x - 2| < δ, then |f(x) - 5| < 1. lim f(x) = 5. Ул y = f(x) 1 х
Evaluate the following limits. 10 lim ( 3 +
Determine the following limits analytically. |lim V5x + 6
The graph of f in the figure has vertical asymptotes at x = 1 and x = 2. Analyze the following limits.a. limx→1- f(x)b. limx→1+ f(x)c. limx→1 f(x)d. limx→2- f(x)e. limx→2+ f(x)f. limx→2 f(x) УА U. iy = f(x) х
Use the graph of g in the figure to find the following values or state that they do not exist.a. g(0)b. limx→0 g(x)c. g(1)d. limx→1g(x) y. 5- 4- 3+ 2- у%3 в() -2 -1 х 2
Explain the Intermediate Value Theorem using words and pictures.
Give the definition ofand interpret it using pictures. lim f(x)
Use a sketch to find the end behavior of f(x) = ln x.
Determine the following limits analytically. lim .2 187? x→1000
Use the graph of f(x) = x/(x2 - 2x - 3)2 to determine limx→-1 f(x) and limx→3 f(x). y = f(x) х 3
Evaluate and lim e*, lim e*, lim e*
Use the graph of h in the figure to find the following values or state that they do not exist.a. h(2)b. limx→2h(x)c. h(4)d. limx→4h(x)e. limx→5h(x) YA 4 y = h(x) 3. 6.
What is the domain of f (x) = ex/x and where is f continuous?
Suppose |f(x) - 5| < 0.1 whenever 0 < x < 5. Find all values of δ > 0 such that |f (x) - 5| < 0.1 whenever 0 < |x - 2| < δ.
Sketch the graph of a function f with all the following properties. lim f(x) x→-2+* lim f(x) х—0 lim_f(x) = 00 = 00 -00 х>-2 lim f(x) = 4 lim_f(x) = 2 f(3) = 1 х>3 x→3+*
Compute the values of f(x) = x + 1/(x - 12) in the following table and use them to determine limx→1 f(x).
Describe the points (if any) at which a rational function fails to be continuous.
What are the potential problems of using a graphing utility to estimate limx→af(x)?
The text describes four cases that arise when examining the end behavior of a rational function f(x) = p(x)/q(x). Describe the end behavior associated with each case.
Explain the meaning of limx→af(x) = L.
What is a horizontal asymptote?
Give the three conditions that must be satisfied by a function to be continuous at a point.
Use a graph to explain the meaning of limx→a f(x) = ∞.
For what values of a does limx→a r(x) = r(a) if r is a rational function?
Use the graph of f in the figure to determine the values of x in the interval (-3, 5) at which f fails to be continuous. Justify your answers using the continuity checklist. УА 5- У 3f(x) 3- х -3 -2 -1 4. 3. 4+
Which one of the following intervals is not symmetric about x = 5?a. (1, 9)b. (4, 6)c. (3, 8)d. (4.5, 5.5)
Determine Г) lim if f(x) → 100,000 and g(x) → 00 as x → 8(х) х 0
Explain the meaning of limx→a+ f(x) = L.
What does it mean for a function to be continuous on an interval?
What is a vertical asymptote?
a. Graphwith a graphing utility. Comment on any inaccuracies in the graph and then sketch an accurate graph of the function.b. Estimateusing the graph in part (a).c. Verify your answer to part (b) by finding the value ofanalytically using the trigonometric identity sin 2θ = 2 sin θ cos θ. sin 20
Does the set {x: 0 < |x - a| < δ} include the point x = a? Explain.
Describe the end behavior of g(x) = e-2x.
We informally describe a function f to be continuous at a if its graph contains no holes or breaks at a. Explain why this is not an adequate definition of continuity.
Consider the function F(x) = f(x)/g(x) with g(a) = 0. Does F necessarily have a vertical asymptote at x = a? Explain your reasoning.
a. Estimateby making a table of values of for values of x approaching π/4. Round your estimate to four digits.b. Use analytic methods to find the value of lim x→/4 cos x – sin x cos 2x cos 2x cos x sin x
State the precise definition of lim f(x) = L.
Describe the end behavior of f(x) = -2x3.
If limx→af(x) = L and limx→a+f(x) = M, where L and M are finite real numbers, then how are L and M related if limx→af(x) exists?
Complete the following sentences.a. A function is continuous from the left at a if _________.b. A function is continuous from the right at a if _________.
Suppose f(x) → 100 and g(x) → 0, with g(x) < 0, as x → 2. Determine limx→2 f(x)/g(x).
Evaluate limx→3- 1/x - 3 and limx→3+ 1/x - 3.
Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).a. Graph the function c = f(t) that gives the cost of renting a snowboard for t days, for 0 ≤ t ≤ 5.b. Evaluatec. Evaluated. Interpret the
Interpret |f (x) - L| < ε in words.
True or false: When limx→af(x) exists, it always equals f(a). Explain.
Suppose f (x) lies in the interval (2, 6). What is the smallest value of ε such that |f(x) - 4| < ε?
Use the graph of f in the figure to find the following values, or state that they do not exist.a. f(-1).b. c. d. e. f(1)f. g. h. i. j. lim_f(x) x→-1¯ lim f(x) r→-1+* x-
Use a graph to explain the meaning of limx→a+ f(x) = - ∞.
Which of the following functions are continuous for all values in their domain? Justify your answers.a. a(t) = altitude of a skydiver t seconds after jumping from a planeb. n(t) = number of quarters needed to park legally in a metered parking space for t minutesc. T(t) = temperature t minutes after
Explain the meaning of 10. lim f(x)
Suppose x lies in the interval (1, 3) with x ≠ 2. Find the smallest positive value of δ such that the inequality 0 < |x - 2| < δ is true.
Explain the meaning of limx→af(x) = L.
Determine whether the following statements are true and give an explanation or counterexample.a. The rational function has vertical asymptotes at x = -1 and x = 1.b. Numerical or graphical methods always produce good estimates of c. The value of if it exists, is found by calculating f (a).d. Ifdoes
Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.a. If limx→a f(x) = L, then f(a) = L.b. If limx→a f(x) = L, then limx→a+ f(x) = L.c. If limx→a f(x) = L and limx→a g(x) = L, then f(a) = g(a).d. The limit
a. Draw a graph to verify that - |x| ≤ x2 ln x2 ≤ |X|, for -1 ≤ x ≤ 1, where x ≠ 0.b. Use the Squeeze Theorem to evaluate limx→0 x2 ln x2.
It can be shown that 1 - x2/6 ≤ sin x/x ≤ 1, for x near 0.a. Illustrate these inequalities with a graph.b. Use these inequalities to evaluate limx→0 sin x/x.
It can be shown that 1 - x2/2 ≤ cos x ≤ 1, for x near 0.a. Illustrate these inequalities with a graph.b. Use these inequalities to evaluate limx→0 cos x.
a. Show that - |x| ≤ x sin 1/x ≤ |x|, for x ≠ 0.b. Illustrate the inequalities in part (a) with a graph.c. Use the Squeeze Theorem to show that limx→0 x sin 1/x = 0.
a. Sketch a graph of y = 3x and carefully draw four secant lines connecting the points P(0, 1) and Q(x, 3x), for x = -2, -1, 1, and 2.b. Find the slope of the line that passes through P(0, 1) and Q(x, 3x), for x ≠ 0.c. Complete the table and make a conjecture about the value of 3* lim х -0.1
a. Sketch a graph of y = 2x and carefully draw three secant lines connecting the points P(0, 1) and Q(x, 2x), for x = -3, -2, and -1.b. Find the slope of the line that passes through P(0, 1) and Q(x, 2x), for x ≠ 0.c. Complete the table and make a conjecture about the value of 2* – 1 lim -1
Evaluate the following limits, where a and b are fixed real numbers. x³ - a lim х
Evaluate the following limits, where a and b are fixed real numbers. V16 + h – 4 lim h-0
Evaluate the following limits, where a and b are fixed real numbers. ? - a? lim a VI - Va
Evaluate the following limits, where a and b are fixed real numbers. х — а lim- х- а Vх - Va
Evaluate the following limits, where a and b are fixed real numbers. 4t lim (6 +t - 12) -3
Evaluate the following limits, where a and b are fixed real numbers. VI - 3- - lim 9 х — 9
Evaluate the following limits, where a and b are fixed real numbers. lim
Evaluate the following limits, where a and b are fixed real numbers. (2x – 1)? – 9 lim
Evaluate the following limits, where a and b are fixed real numbers. (x + b)7+(x + b)10 4(x + b) lim
Evaluate the following limits, where a and b are fixed real numbers. (x – b)50 – x + b - x + b lim
Evaluate the following limits, where a and b are fixed real numbers. 312 – 7t + 2 2 t lim
Evaluate the following limits, where a and b are fixed real numbers. x? 16 х lim 4 4 - xr
Evaluate the following limits, where a and b are fixed real numbers.limx→3 x2 - 2x - 3/x - 3
Evaluate the following limits, where a and b are fixed real numbers.limx→1 x2 - 1/x - 1
Show that limx→a |x| = |a|, for any real number.
a. Evaluate limx→3- √x - 3/2 - x.b. Explain why limx→3+ √x - 3/2 - x does not exist.
a. Evaluate limx→2+ √x - 2.b. Explain why limx→2- √x - 2 does not exist.
LetCompute the following limits or state that they do not exist.a. limx→-5- f(x)b. limx→-5+ f(x)c. limx→-5 f(x)d. limx→5- f(x)e. limx→5+ f(x)f. limx→5 f(x) if x s -5 f(x) = V25 — х2 if -5
LetCompute the following limits or state that they do not exist.a. limx→-1- f(x)b. limx→-1+ f(x)c. limx→-1 f(x) Sx² + 1 ifx < -1 | VI + 1 ifx 2 -1. f(x) =
Evaluate the following limits. 3 lim h-0 V16 + 3h + 4
Evaluate the following limits. -5x lim -3 V4x - 3
Evaluate the following limits.limx→2 (x2 - x)5
Evaluate the following limits. 3b lim b2 V4b + I – 1
Evaluate the following limits. lim Vr? +3 – 10
Evaluate the following limits.limx→1 5x2 + 6x + 1/8x - 4
Evaluate the following limits.limt→-2 (t2 + 5t + 7)
Evaluate the following limits.limx→1 (2x3 - 3x2 + 4x + 5)
Assume limx→1 f(x) = 8, limx→1 g(x) = 3, and limx→1 h(x) = 2. Compute the following limits and state the limit laws used to justify your computations. lim Vf(x)g(x) + 3
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