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mathematics
calculus early transcendentals

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

Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.
Explain why the slope of a secant line can be interpreted as an average rate of change.
Use definition (1) for the slope of a tangent line to explain how slopes of secant lines approach the slope of the tangent line at a point.Definition (1) f(x) – f(a) lim х — а (1) Mtan ха
Give a formal proof that lim —2 (х — 2)4
a. Assume |f (x)| ≤ L for all x near a andGive a formal proof thatb. Find a function f for whichWhy doesn’t this violate the result stated in (a)?c. The Heaviside function is defined asExplain why 0. lim g(x) lim (f(x)g(x)) = 0.
Give a formal proof that x2 lim x→5 X – 5 25 10.
Give a formal proof that lim (5x – 2) = 3. x→1
The amount of an antibiotic (in mg) in the blood t hours after an intravenous line is opened is given bym(t) = 100(e-0.1t - e-0.3t).a. Use the Intermediate Value Theorem to show the amount of drug is 30 mg at some time in the interval [0, 5] and again at some time in the interval [5, 15].b.
a. Use the Intermediate Value Theorem to show that the equation x5 + 7x + 5 = 0 has a solution in the interval (-1, 0).b. Estimate a solution to x5 + 7x + 5 = 0 in (-1, 0) using a root finder.
Sketch the graph of a function that is continuous on (0, 1) and continuous on (1, 2) but is not continuous on (0, 2).
a. Is h(x) = √x2 - 9 left-continuous at x = 3? Explain.b. Is h(x) = √x2 - 9 right-continuous at x = 3? Explain
Determine values of the constants a and b for which g is continuous at x = 1. 5х — 2 if x < 1 8(x) = if x = 1 if x > 1. ax² + bx .2 ах
Find the intervals on which the following functions are continuous. Specify right- or left continuity at the endpoints.g(x) = cos ex
Find the intervals on which the following functions are continuous. Specify right- or left continuity at the endpoints. 2x h(x) - 25x
Find the intervals on which the following functions are continuous. Specify right- or left continuity at the endpoints.g(x) = e√x - 2
Find the intervals on which the following functions are continuous. Specify right- or left continuity at the endpoints.f(x) = √x2 - 5
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answers. .2 16 if x # 4 ; a = 4 g(x) if x = 4
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answers.h(x) = √x2 - 9; a = 3
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answers. .2 16 if x + 4 8(x) ; a = 4| if x = 4 9.
Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answers. f(x) a = 5 x – 5'
a. Analyzefor each function.b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. lim f(x) and lim f(x) f(x) x(х + 2)5 3x2 – 4x
a. Analyzefor each function.b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. lim f(x) and lim f(x) 1 + x – 2r? – x3 f(x)
a. Analyzefor each function.b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. lim f(x) and lim f(x) .2 9x2 + 4 f(x) (2x – 1)² 1)2
a. Analyzefor each function.b. Determine whether f has a slant asymptote. If so, write the equation of the slant asymptote. lim f(x) and lim f(x) f(x) = 3x? + 2x – 1 4x + 1
Find all vertical and horizontal asymptotes of the following functions. 2x2 + 6 f(x) 2x2 + 3x – 2
Find all vertical and horizontal asymptotes of the following functions.f(x) = 1/tan-1 x
Determine the end behavior of the following functions.f(x) = 1/ln x2
Determine the end behavior of the following functions.f(x) = 1 - e-2x
Analyze |x – 1 |x – 1 lim r→1+ V x – 3 and lim x→1¯ V x – 3
Suppose f is continuous at a and assume f(a) > 0. Show that there is a positive number δ > 0 for which f(x) > 0 for all values of x in (a - δ, a + δ). (In other words, f is positive for all values of x in the domain of f and in some interval containing a.)
Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write if for some ε > 0, there is no value of δ > 0 satisfying the condition|f(x) - L| < ε whenever 0 < |x - a| < δLetProve that does not
Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write if for some ε > 0, there is no value of δ > 0 satisfying the condition|f(x) - L| < ε whenever 0 < |x - a| < δProve that does not
Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write if for some ε > 0, there is no value of δ > 0 satisfying the condition|f(x) - L| < ε whenever 0 < |x - a| < δFor the following
Suppose f is defined for all values of x near a, except possibly at a. Assume for any integer N > 0 there is another integer M > 0 such that |f(x) - L| < 1/N whenever |x - a| < 1/M. Prove that using the precise definition of a limit. lim f(x) = L
Assume the functions f, g, and h satisfy the inequality f(x) ≤ g(x) ≤ h(x) for all values of x near a, except possibly at a. Prove that if then lim f(x) = lim h(x) = x→a |lim g(x) = L.
We write if for any positive number M there is a corresponding N > 0 such thatf(x) > M whenever x > N.Use this definition to prove the following statements. lim f(x) = 00 x- .2 х lim х 00 X- х
We write if for any positive number M there is a corresponding N > 0 such thatf(x) > M whenever x > N.Use this definition to prove the following statements. lim f(x) = 00 x- lim x→o 100 ||
The limit at infinity means that for any ε > 0 there exists N > 0 such thatUse this definition to prove he following statements. lim f(x) = L х- \f(x) – L| < ɛ whenever x > N. $\f(x) – L| < ɛ whenever x style=$
The limit at infinity means that for any ε > 0 there exists N > 0 such thatUse this definition to prove he following statements. lim f(x) = L х- \f(x) – L| < ɛ whenever x > N. $\f(x) – L| < ɛ whenever x style=$
We write if forany negative number M there exists δ > 0 such thatUse this definition to prove the following statements. lim f(x) 00 x→a f(x) < M whenever 0 < |x – a| < 8.
We write if forany negative number M there exists δ > 0 such thatUse this definition to prove the following statements. lim f(x) 00 x→a f(x) < M whenever 0 < |x – a| < 8.
Use the definitions given in Exercise 45 to prove the following infinite limits.Data from Exercise 45We write if for any negative number N there exists δ > 0 such that 1 lim x→1¯ 1 – X lim f(x) -00 ||
Use the definitions given in Exercise 45 to prove the following infinite limits.Data from Exercise 45We write if for any negative number N there exists δ > 0 such that lim x→I* 1 – x lim f(x) -00 ||
We write if for any negative number N there exists δ > 0 such thata. Write an analogous formal definition for b. Write an analogous formal definition for c. Write an analogous formal definition for lim f(x) -00 || f(x) < N whenever 0
Prove the following statements to establish the fact that a.
Use the following definitions. Assume f exists for all x near a with x > a. We say the limit of f(x) as x approaches a from the right of a is L and write, if for any ε > 0 there exists δ > 0 such thatAssume f exists for all x near a with x < a. We say the limit of f(x) as x approaches
Use the following definitions. Assume f exists for all x near a with x > a. We say the limit of f(x) as x approaches a from the right of a is L and write, if for any ε > 0 there exists δ > 0 such thatAssume f exists for all x near a with x < a. We say the limit of f(x) as x approaches
Use the following definitions. Assume f exists for all x near a with x > a. We say the limit of f(x) as x approaches a from the right of a is L and write, if for any ε > 0 there exists δ > 0 such thatAssume f exists for all x near a with x < a. We say the limit of f(x) as x approaches
Use the following definitions. Assume f exists for all x near a with x > a. We say the limit of f(x) as x approaches a from the right of a is L and write, if for any ε > 0 there exists δ > 0 such thatAssume f exists for all x near a with x < a. We say the limit of f(x) as x approaches
Use the following definitions. Assume f exists for all x near a with x > a. We say the limit of f(x) as x approaches a from the right of a is L and write, if for any ε > 0 there exists δ > 0 such thatAssume f exists for all x near a with x < a. We say the limit of f(x) as x approaches
Use the definition of a limit to prove the following results. lim .2 x-5 x 25
Use the definition of a limit to prove the following results.Multiply the numerator and denominator by √x + 2. х — 4 lim |х—4 Vх — 2 = 4
Use the definition of a limit to prove the following results.As x→3, eventually the distance between x and 3 is less than 1. Start by assuming |x - 3| < 1 and show1/|x| < 1/2. lim 3 x→3 X
Let f(x) = x2 - 2x + 3.a. For ε = 0.25, find the largest value of δ > 0 satisfying the statement|f(x) - 2| < ε whenever 0 < |x - 1| < δ.b. Verify thatas follows. For any ε > 0, find the largest value of δ > 0 satisfying the statement|f(x) - 2| < ε whenever 0 < |x - 1|
Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume a. For a given ε > 0, there is one value of δ > 0 for which |f(x) - L| < ε whenever 0 < | x - a| < δ.b. The limit means that given an
Use the precise definition of infinite limits to prove the following limits. lim sin x .4
Use the precise definition of infinite limits to prove the following limits. lim 00
Use the precise definition of infinite limits to prove the following limits. 1 = 00 lim x→-1 (x + 1)ª
Use the precise definition of infinite limits to prove the following limits. lim x→4 (x - 4)? ||
If f(x) = mx + b, then for constants m ≠ 0 and b (the case m = 0). For a given ε > 0, let δ = ε/|m|.) Explain why this result implies that linear functions are continuous. — та + b lim f(x)
Give proofs of the following theorems.a. for any constant c.b. for any constant a. | lim c = c х—а lim x = a х—а
Suppose Prove that where c is a constant. lim f(x) = L. х- х—а ►a lim [cf(x)] = cL,
Suppose Prove that lim f(x) = L and lim g(x) = M. lim (f(x) – g(x)) = L – M. х—а
Use the precise definition of a limit to prove the following limits.Use the identity √x2 = |x|. lim (x – 3)² = 0 x→3
Use the precise definition of a limit to prove the following limits.Use the identity √x2 = |x|. .2 |X'
Use the precise definition of a limit to prove the following limits. 7х + 12 x² lim х—3 -1 х — 3
Use the precise definition of a limit to prove the following limits. х* lim .2 16 х>4 х — 4
Use the precise definition of a limit to prove the following limits. lim (-2x + 8) = 2 х—3
Use the precise definition of a limit to prove the following limits. lim (8x + 5) = 13 |х—1
Letand note thatFor each value of ε, use a graphing utility to find all values of δ > 0 such that |f(x) - 2| < ε whenever 0 < |x - 3| < δ.a. ε = 1/2b. ε = 1/4c. For any ε > 0, make a conjecture about the value of δ that satisfies the preceding inequality. zx + 1 글x +
Let and note that For each value of ε, use a graphing utility to find all values of δ > 0 such that |f(x) - 4| < ε whenever 0 < |x - 1| < δ.a. ε = 2b. ε = 1c. For any ε > 0, make a conjecture about the value of δ that satisfies the preceding inequality. 2x2 – 2 х —
a. Use the identity sin (a + h) = sin a cos h + cos a sin h with the fact that to prove that thereby establishing that sin x is continuous for all x. (Hint: Let h = x - a so that x = a + h and note that h→ 0 as x→ a.)b. Use the identity cos 1a + h2 = cos a cos h - sin a sin h with the fact that
a. Find functions f and g such that each function is continuous at 0 but f º g is not continuous at 0.b. Explain why examples satisfying part (a) do not contradict Theorem 2.12.
Prove Theorem 2.11: If g is continuous at a and f is continuous at g(a), then the composition f º g is continuous at a. (Hint: Write the definition of continuity for f and g separately; then combine them to form the definition of continuity for f º g.)
Classify the discontinuities in the following functions at the given points. - 4x2 + 4x O and x = 1 h(x) ;x = x(х — 1)
Classify the discontinuities in the following functions at the given points. |x – 2| ; х —D 2 F(x) х — 2
Do removable discontinuities exist?a. Does the function f(x) = x sin (1/x) have a removable discontinuity at x = 0?b. Does the function g(x) = sin (1/x) have a removable discontinuity at x = 0?
Show that the following functions have a removable discontinuity at the given point. x² – 1 .2 if x + 1 ; x = 1 g(x) if x = 1 3
Show that the following functions have a removable discontinuity at the given point. x² – 7x + 10 f(x) =
Is the discontinuity at a in graph (c) removable? Explain.
The discontinuities in graphs (a) and (b) are removable discontinuities because they disappear if we define or redefine f at a so that The function in graph (c) has a jump discontinuity because left and right limits exist at a but are unequal. The discontinuity in graph (d) is an infinite
Leta. Write a formula for |g(x)|.b. Is g continuous at x = 0? Explain.c. Is g continuous at x = 0? Explain.d. For any function f, if |f| is continuous at a, does it necessarily follow that f is continuous at a? Explain. if x > 0 8(x) : if x < 0. -1
A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day the monk made the return walk to the valley, leaving the temple at dawn, walking the same
Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 a.m. on a Friday morning. On Sunday morning, you leave the lake at 7 a.m. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f 1t2 be your distance from the car t hours
Assume you invest $250 at the end of each year for 10 years at an annual interest rate of r. The amount of money in your account after 10 years isAssume your goal is to have $3500 in your account after 10 years.a. Use the Intermediate Value Theorem to show that there is an interest rate r in the
Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0 ≤ t ≤ 60. yA 1.25 1.00 y = c(t) 0.75 0.50 0.25 O + 60 15 30 45 Time (min) Cost (dollars)
Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots.70x3 - 87x2 + 32x - 3 = 0; (0, 1)
Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots.x3 + 10x2 - 100x + 50 = 0; (-20, 10)
Then give the horizontal and vertical asymptotes of f. Plot f to verify your results. 2e* + 10e f(x) Analyze lim f(x), lim f(x), and lim f(x).
Letand Then give the horizontal and vertical asymptotes of f. Plot f to verify your results. 2e* + 5e3x Analyze lim f(x), lim f(x), lim f(x), X -00 F(x) e2x e3x x→0+*
Leta. Determine the value of a for which g is continuous from the left at 1.b. Determine the value of a for which g is continuous from the right at 1.c. Is there a value of a for which g is continuous at 1? Explain x² + x .2 if x < 1 if x = 1 8(x) 3x + 5 if x > 1.
Determine the value of the constant a for which the functionis continuous at -1. x2 + 3x + 2 if x + -1 x + 1 f(x) if x = -1
a. Sketch the graph of a function that is not continuous at 1, but is defined at 1.b. Sketch the graph of a function that is not continuous at 1, but has a limit at 1.
Graph the function f(x) = sin x/x using a graphing window of [-π, π] × [0, 2].a. Sketch a copy of the graph obtained with your graphing device and describe any inaccuracies appearing in the graph.b. Sketch an accurate graph of the function. Is f continuous at 0?c. What is the value of sin
The graph of the sawtooth function is the greatest integer function or floor function, was obtained using a graphing utility (see figure). Identify any inaccuracies appearing in the graph and then plot an accurate graph by hand. y = x - [x], where [x]]
Evaluate the following limits or state that they do not exist. lim x→0+ In x
Evaluate the following limits or state that they do not exist. lim x→1¯ In x
Evaluate the following limits or state that they do not exist. cos t lim t→0 e3t
Evaluate the following limits or state that they do not exist. tan lim х х х-
Evaluate the following limits or state that they do not exist. 1 - cos? x lim x→0+ sin x

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