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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Refer to the graphs to explain why the equation sinh x = t has a unique solution for every t and why cosh x = t has two solutions for every t > 1.
A homework problem asks for a sketch of the graph of the inverse of ƒ(x) = x + cos x. Frank, after trying but failing to find a formula for ƒ−1(x), says it’s impossible to graph the inverse.
Verify that ƒ(x) = x3 + 3 and g(x) = (x − 3)1/3 are inverses by showing that ƒ(g(x)) = x and g(ƒ(x)) = x.
Find the equation of the tangent line at the point indicated.y = 4ex, x0 = 0
Which relation between x and u yields √16 + x2 = 4 √1 + u2?
Calculate without using a calculator.log2(25/3)
In one year, you will be paid $1. Will the PV increase or decrease if the interest rate goes up at the start of the year?
What property of the function ƒ(x) = ex allows us to say lim es(x) x→a lim g(x) q = ex-a
Show that g(x) = x/x − 1 is equal to its inverse on the domain {x : x ≠ −1}.
Evaluate the limit, using L’Hôpital’s Rule where it applies. √x + 1-2 lim x3 x³-7x-6
A quantity P obeys the exponential growth law P(t) = Cekt (t in years). Find the formula for P(t), assuming that P(0) = 100 and that it takes 5 years for P to double.
What is the slope of the line obtained by reflecting the line y = x/2 through the line y = x?
Find the equation of the tangent line at the point indicated.y = e4x, x0 = 0
Does ƒ(x) = x−1 have an antiderivative for x < 0? If so, describe one.
Calculate without using a calculator.log2(85/3)
For y = log10 x, if y increases by 2, then (choose the correct answer):(a) x increases by 20(b) x decreases by 20(c) x increases by a factor of 2(d) x increases by a factor of 100
Describe the graphical interpretation of the relation g'(x) = 1/ ƒ'(g(x)), where ƒ and g are inverses of each other.
A 10-kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.
Evaluate the limit, using L’Hôpital’s Rule where it applies. lim x0x² sin 4x + 3x + 1
Prove the addition formula for cosh x.
The escape velocity from a planet of radius R is where G is the universal gravitational constant and M is the mass. Find the inverse of v expressing R in terms of v. v(R): = 2GM R
Suppose that P = (2, 4) lies on the graph of ƒ and that the slope of the tangent line through P is m = 3. Assuming that ƒ−1 exists, what is the slope of the tangent line to the graph of
Find the equation of the tangent line at the point indicated.y = ex+2, x0 = −1
Suppose that g is the inverse of ƒ. Match the functions (a)–(d) with their inverses (i)–(iv). (a) f(x) + 1 (b) f(x + 1) (c) 4f(x) (d) f(4x) (i) g(x)/4 (ii) g(x/4) (iii) g(x - 1) (iv) g(x) - 1
What is the slope of the tangent line to y = 4x at x = 0?
Calculate without using a calculator.log64 4
Assuming that population growth is approximately exponential, which of the following two sets of data is most likely to represent the population (in millions) of a city over a 5-year period? Year
Evaluate the limit, using L’Hôpital’s Rule where it applies. lim x-0 sin x - x
Use the addition formulas to prove sinh(2x) = 2 cosh x sinh x cosh(2x) = cosh2 x + sinh2 x
Show that the power law relationship P(Q) = kQr, for Q ≥ 0 and r ≠ 0, has an inverse that is also a power law, Q(P) = mPs, where m = k−1/r and s = 1/r.
Find the equation of the tangent line at the point indicated.y = ex2 , x0 = 1
What is the rate of change of y = ln x at x = 10?
Calculate without using a calculator.log7(492)
Sam was 28 inches tall on her first birthday, 50 inches tall on her eighth, and 62 inches tall on her 14th.(a) Let t represent Sam’s age in years, and let h represent her height in inches.
Find g'(8) where g is the inverse of a differentiable function ƒ such that ƒ(−1) = 8 and ƒ(−1) = 12.
Evaluate the limit, using L’Hôpital’s Rule where it applies. lim x→0 cos 2x 1 sin 5x
The volume V of a cone that has height equal to its radius r is given by V(r) = 1/3 πr3. Find the inverse of V(r), expressing r as a function of V.
Calculate without using a calculator.log8 2 + log4 2
Suppose that ƒ(g(x)) = ex2, where g'(1) = 2 and g'(1) = 4. Find ƒ'(2).
Evaluate the limit, using L’Hôpital’s Rule where it applies. lim X-0 cos.x - sin² x sin x
Suppose $500 is deposited into an account paying interest at a rate of 7%, continuously compounded. Find a formula for the value of the account at time t. What is the value of the account after 3
Calculate without using a calculator. 5 log25 30 +10g25 6
The surface area S of a sphere of radius r is given by S (r) = 4πr2. Explain why, in the given context, S (r) has an inverse function. Find the inverse of S (r), expressing r as a function of S.
Show that if ƒ is a function satisfying ƒ'(x) = ƒ(x)2, then its inverse g satisfies g'(x) = x−2.
Use L’Hôpital’s Rule to evaluate the limit. 9x + 4 2x lim x-00 3 -
How long will it take for $4000 to double in value if it is deposited in an account bearing 7% interest, continuously compounded?
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1.ƒ(x) = 4 − x
Use L’Hôpital’s Rule to evaluate the limit. lim xsin - X→-00
Calculate without using a calculator.log4 48 − log412
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1. f(x) = 1 x + 1
How much must one invest today in order to receive $20,000 after 5 years if interest is compounded continuously at the rate r = 9%?
Use L’Hôpital’s Rule to evaluate the limit. In x lim X→∞0x1/2
Calculate without using a calculator.ln(√e · e7/5)
An investment increases in value at a continuously compounded rate of 9%. How large must the initial investment be in order to build up a value of $50,000 over a 7-year period?
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1. f(x) = 1 7x - 3
Calculate without using a calculator.ln(e3) + ln(e4)
Compute the PV of $5000 received in 3 years if the interest rate is (a) 6% (b) 11%. What is the PV in these two cases if the sum is instead received in 5 years?
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1.ƒ(z) = z3
Use L’Hôpital’s Rule to evaluate the limit. 1² lim X→∞0 ex x-
For each function shown in Figure 15, sketch the graph of the inverse (restrict the function’s domain if necessary). 7 (A) -X (D) (B) -X (C) htt (E) (F) -X -X
The decibel level for the intensity of a sound is a logarithmic scale defined by D = 10 log10 I + 120, where I is the intensity of the sound in watts per square meter.(a) Express I as a function of
Simplify by referring to the appropriate triangle or trigonometric identity.tan(cos−1 x)
Write as the natural log of a single expression:(a) 2 ln5 + 3 ln4 (b) 5 ln(x1/2) + ln(9x)
Find the PV of an income stream that pays out continuously at a rate R(t) = $5000e0.1t/year for 7 years, assuming r = 0.05.
Calculate without using a calculator.83log8(2)
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1. f(x) = 1 V.x² + 1
Find the PV of an investment that pays out continuously at a rate of $800/year for 5 years, assuming r = 0.08.
Calculate without using a calculator.7log7(29)
The Palermo Technical Impact Hazard Scale P is used to quantify the risk associated with the impact of an asteroid colliding with the earth:where pi is the probability of impact, T is the number of
Show that ƒ(x) = (x2 + 1)−1 is one-to-one on (−∞, 0], and find a formula for ƒ−1 for this domain of ƒ.
Refer to the appropriate triangle or trigonometric identity to compute the given value.tan (sin−1 0.8)
Let ƒ(x) = x2 − 2x. Determine a domain on which ƒ−1 exists, and find a formula for ƒ−1 for this domain of ƒ.
A ball is launched straight up in the air and is acted on by air resistance and gravity as in Example 8. The function H gives the maximum height that the ball attains as a function of the
Refer to the appropriate triangle or trigonometric identity to compute the given value.cos (cot−1 1)
Show that the inverse of ƒ(x) = e−x exists (without finding it explicitly). What is the domain of ƒ−1?
A ball with a mass of 500 g is launched upward with an initial velocity of 30 m/s, and H(K) 15k - 2.45 In(300k 49 k² + 1)
Refer to the appropriate triangle or trigonometric identity to compute the given value.cot(csc−1 2)
Find the inverse g of ƒ(x) = √x2 + 9 with domain x ≥ 0 and calculate g'(x) in two ways: using Theorem 2 and by direct calculation. THEOREM 2 Derivative of the Inverse Assume that f is
The Beer–Lambert Law is used in spectroscopy to determine the molar absorptivity α > 0 or the concentration c > 0 of a compound dissolved in a solution at low concentrations (Figure 1). The
Show, by producing a counterexample, that ln(ab) is not equal to (ln a)(ln b).
Refer to the appropriate triangle or trigonometric identity to compute the given value.tan (sec−1(−2))
Let g be the inverse of ƒ(x) = x3 + 1. Find a formula for g(x) and calculate g'(x) in two ways: using Theorem 2 and then by direct calculation THEOREM 2 Derivative of the Inverse Assume that f is
Consider the logistic world population model from Example 2: (a) Determine the time t when the population is half of the expected long-term limit of 17.4 billion.(b) There is a single point of
What is b if (logb'x)= 1/3x?
Use Theorem 2 to calculate g'(x), where g is the inverse of ƒ.ƒ(x) = 7x + 6 THEOREM 2 Derivative of the Inverse Assume that f is differentiable and one-to- one with inverse g(x) = f(x). If b
Refer to the appropriate triangle or trigonometric identity to compute the given value.cot (tan−1 20)
Consider the general logistic function P(t) = M/1 + Ae−kt , with A, M, and k all positive. Show that Make-kt MAK-et (Ae-kt - 1) and P' (t) = (1 + Ae-kt)2 (1 + Ae-kt)3 (b) lim P(t) = 0 and lim P(t)
Use Theorem 2 to calculate g'(x), where g is the inverse of ƒ.ƒ(x) = √3 − x THEOREM 2 Derivative of the Inverse Assume that f is differentiable and one-to- one with inverse g(x) = f(x). If b
Refer to the appropriate triangle or trigonometric identity to compute the given value.sin (csc−1 20)
Use Theorem 2 to calculate g'(x), where g is the inverse of ƒ.ƒ(x) = x−5 THEOREM 2 Derivative of the Inverse Assume that f is differentiable and one-to- one with inverse g(x) = f(x). If b belongs
Compute the derivative at the point indicated without using a calculator. y = sin ¹x, x =
Use Theorem 2 to calculate g'(x), where g is the inverse of ƒ.ƒ(x) = 4x3 − 1 THEOREM 2 Derivative of the Inverse Assume that f is differentiable and one-to- one with inverse g(x) = f(x). If b
Compute the derivative at the point indicated without using a calculator. y = tan ¹x, x = 1/2
A company can earn additional profits of $500,000/year for 5 years by investing $2 million to upgrade its factory. Is the investment worthwhile if the interest rate is 6%? (Assume the savings are
Compute the derivative at the point indicated without using a calculator.y = sec−1 x, x = 4
Use Theorem 2 to calculate g'(x), where g is the inverse of ƒ. THEOREM 2 Derivative of the Inverse Assume that f is differentiable and one-to- one with inverse g(x) = f(x). If b belongs to the
A new computer system costing $25,000 will reduce labor costs by $7000/year for 5 years.(a) Is it a good investment if r = 8%?(b) How much money will the company actually save?
Compute the derivative at the point indicated without using a calculator.y = arccos(4x), x = 1/5
Use Theorem 2 to calculate g'(x), where g is the inverse of ƒ.ƒ(x) = 2 + x−1 THEOREM 2 Derivative of the Inverse Assume that f is differentiable and one-to- one with inverse g(x) = f(x). If b
Find g'(−1/2), where g is the inverse of f(x) = p³ 2+1 1-²

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