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mathematics
applied calculus

Questions and Answers of Applied Calculus

Decide whether the expression is a number or a family of functions. (Assume f(x) is a function.) -5 ľ 3 2f(x) dx
Figure 6.8 shows the rate of change of the concentration of adrenaline, in micrograms per milliliter per minute, in a person’s body. Sketch a graph of the concentration of adrenaline, in micrograms
Use the Fundamental Theorem to evaluate the definite integral exactly. -2 5t³ dt 3
Find the integrals in Problems. Check your answers by differentiation. [Se³i+2di dt
Find the integrals in Problems. 1+7 1. √5-t dt
A small business expects an income stream of $5000 per year for a four-year period.(a) Find the present value of the business if the annual interest rate, compounded continuously, is(i) 3% (ii)
Find the integrals in Problem. In x x² dx
Find the integrals in Problems. Check your answers by differentiation. хе-х² dx [xe
Decide whether the expression is a number or a family of functions. (Assume f(x) is a function.) /5 5 dx
Use the Fundamental Theorem to evaluate the definite integral exactly. 3 J 6x² dx
A small business, Company A, believes it can generate an income stream of $120,000 per year for the next 20 years. A larger firm, Company B, offers to purchase Company A for a lump sum of $2 million
Decide whether the expression is a number or a family of functions. (Assume f(x) is a function.) [ f(5) dx
Problems show the derivative f' of f.(a) Where is f increasing and where is f decreasing? What are the x-coordinates of the local maxima and minima of f?(b) Sketch a possible graph for f. (You
For a product, the supply curve is given by q = 10p−30 and the demand curve by q = −2p + 30, where p is in dollars and q is in millions.(a) What is the equilibrium price and quantity?(b) Find the
(a) Estimate the equilibrium price and quantity for the supply and demand curves in Figure 6.42.(b) Estimate the consumer and producer surplus.(c) The price is set artificially low at p− = 4
Find the integrals in Problems. Check your answers by differentiation. Je e-0.11+ dt
Use the Fundamental Theorem to evaluate the definite integral exactly. [₁0³² + 3 (² + y²) dy
Find the integrals in Problems. [₁ t sint dt
The Hershey Company is the largest US producer of chocolate. Between 2011 and 2014, Hershey generated net sales at a rate approximated by 5.3 + 0.42t billion dollars per year, where t is the time in
Decide whether the expression is a number or a family of functions. (Assume f(x) is a function.) x+1 dx
Problems show the derivative f' of f.(a) Where is f increasing and where is f decreasing? What are the x-coordinates of the local maxima and minima of f?(b) Sketch a possible graph for f. (You
Find the integrals in Problem. (0+1) sin(0+1) de
German TV and radio users pay an annual license fee of 200 euros. In 2008, the German poet Friedrich Schiller, who died in 1805, was sent reminders to pay his license fee. (The reminders were sent to
Find the integrals in Problems. Check your answers by differentiation. " 100e-0.21 dt
Use the Fundamental Theorem to evaluate the definite integral exactly. S' 0 e-0.21 dt
Use the Fundamental Theorem to evaluate the definite integral exactly. xp Z
The derivative F'(t) is graphed in Figure 6.9. Given that F(0) = 5, calculate F(t) for t = 1, 2, 3, 4, 5. 0 -1 -2 1 2 F' (t) 3 Figure 6.9 4 5
Rent controls on apartments are an example of price controls on a commodity. They keep the price artificially low (below the equilibrium price). Sketch a graph of supply and demand curves, and label
Sketch two functions F such that F' = f. In one case let F(0) = 0 and in the other, let F(0) = 1. 1 -1 f(x)
A recently installed machine earns the company revenue at a continuous rate of 60,000t+45,000 dollars per year during the first six months of operation and at the continuous rate of 75,000 dollars
Your company needs $500,000 in two years’ time for renovations and can earn 9% interest on investments.(a) What is the present value of the renovations?(b) If your company deposits money
Find an antiderivative.f(t) = 5t
The supply and demand curves have equations p = S(q) and p = D(q), respectively, with equilibrium at (q∗, p∗).Using Riemann sums, explain the economic significance of ∫q∗0 S(q) dq to the
Find ∫21 ln x dx numerically. Find ∫21 ln x dx using antiderivatives. Check that your answers agree.
Sales of Version 6.0 of a computer software package start out high and decrease exponentially. At time t, in years, the sales are s(t) = 50,000e−t dollars per year. After two years, Version 7.0 of
(a) Estimate ∫40 f(x) dx for f(x) in Figure 6.10.(b) Let F(x) be an antiderivative of f(x). Is F(x) increasing or decreasing on the interval 0 ≤ x ≤ 4?(c) If F(0) = 20, what is F(4)? 2 1 f(x) 2
Use the Fundamental Theorem to evaluate the definite integral exactly. 1 -1 cost dt
Sketch two functions F such that F' = f. In one case let F(0) = 0 and in the other, let F(0) = 1. 1 -1 f(x) 1 x
Find the integrals in Problems. Check your answers by differentiation. [x(x x(x² - 4)7/² dx
A company is expected to earn $50,000 a year, at a continuous rate, for 8 years. You can invest the earnings at an interest rate of 7%, compounded continuously. You have the chance to buy the rights
Evaluate the integrals in Problem both exactly [e.g. ln(3π)] and numerically [e.g. ln(3π) ≈ 2.243]. zp z_az 10 0.
Find the integrals in Problem. Check your answers by differentiation. [x(x² + 3)² dx
Use the Fundamental Theorem to evaluate the definite integral exactly. [** (sint + cost) dt
Evaluate the integrals in Problems both exactly [e.g. ln(3π)] and numerically [e.g. ln(3π) ≈ 2.243]. 5 Int dt 1
Sketch two functions F such that F' = f. In one case let F(0) = 0 and in the other, let F(0) = 1. -1 f(x) X
Find an antiderivative.f(x) = x2
The supply and demand curves have equations p = S(q) and p = D(q), respectively, with equilibrium at (q∗, p∗).Using Riemann sums, give an interpretation of producer surplus, ∫q∗0 (p∗ −
Sketch two functions F such that F' = f. In one case let F(0) = 0 and in the other, let F(0) = 1. -1 f(x) 1 X
Find the integrals in Problems. Check your answers by differentiation. 1 (3x + 1)² J dx
Evaluate the integrals in Problem both exactly [e.g. ln(3π)] and numerically [e.g. ln(3π) ≈ 2.243]. 5 In(1+t) dt
Use the Fundamental Theorem to evaluate the definite integral exactly. 3 e0.05t dt
Use the Fundamental Theorem to evaluate the definite integral exactly. 4 √xdx
Find an antiderivative.g(t) = t2 + t
Sketch two functions F such that F' = f. In one case let F(0) = 0 and in the other, let F(0) = 1. 1 -1 1 f(x) X
Find the integrals in Problem. Check your answers by differentiation. 4x³ x4 +1 xp. dx
Intel Corporation is a leading manufacturer of integrated circuits. In 2011, Intel generated profits at a continuous rate of 34.6 billion dollars per year based on a total revenue of 54 billion
Evaluate the integrals in Problems both exactly [e.g. ln(3π)] and numerically [e.g. ln(3π) ≈ 2.243]. S tlnt dt
Find the integrals in Problem. Check your answers by differentiation. [12x² cos(x³)dx
Find an antiderivative.f(x) = 5
Harley-Davidson Inc. manufactures motorcycles. During the years following 2013 (the company’s 110th anniversary year), the company’s net revenue can be approximated by 5.9+0.3t billion dollars
Find an antiderivative.f(x) = x4
McDonald’s Corporation licenses and operates over 30,000 fast-food restaurants throughout the world. Between 2005 and 2008, McDonald’s generated revenue at continuous rates between 17.9 and 22.8
Evaluate the integrals in Problems both exactly [e.g. ln(3π)] and numerically [e.g. ln(3π) ≈ 2.243]. √3 x cos x dx
Find the area under the graph of f(t) for 0 ≤ t ≤ 5 using the Fundamental Theorem of Calculus.Compare your answer with what you get using areas of triangles.f(t) = 6t
Find the area under the graph of f(t) for 0 ≤ t ≤ 5 using the Fundamental Theorem of Calculus.Compare your answer with what you get using areas of triangles.f(t) = 10 − 2t
Find an antiderivative.g(t) = t7 + t3
Find the integrals in Problem. Check your answers by differentiation. [ (2t - 7)73 dt
Figure 6.11 shows the derivative F' of a function F. If F(20) = 150, estimate the maximum value attained by F. 20 10 -10 20 F'(x) Figure 6.11 40 50 60
Your company is considering buying new production machinery. You want to know how long it will take for the machinery to pay for itself; that is, you want to find the length of time over which the
Use integration by parts twice to evaluate the integral. [ (In t)² dt
Find the area under the graph of f(t) for 0 ≤ t ≤ 5 using the Fundamental Theorem of Calculus.Compare your answer with what you get using areas of triangles.f(t) = t
Find the integrals in Problem. Check your answers by differentiation. [(x² + 3)² dx
Find an antiderivative.f(q) = 5q2
Table 6.2 shows the monthly change in water stored in Lake Sonoma, California, from March through November 2014. The change is measured in acre-feet per month.1 On March 1, the water stored was
An oil company discovered an oil reserve of 100 million barrels. For time t > 0, in years, the company’s extraction plan is a linear declining function of time as follows:q(t) = a − bt,where
Use integration by parts twice to evaluate the integral. test dt
Find the area under the graph of f(t) for 0 ≤ t ≤ 5 using the Fundamental Theorem of Calculus.Compare your answer with what you get using areas of triangles.f(t) = −2t
Find an antiderivative.g(x) = 6x3 + 4
Find the integrals in Problem. Check your answers by differentiation. | (1 + y) dy
The value of some good wine increases with age. Thus, if you are a wine dealer, you have the problem of deciding whether to sell your wine now, at a price of $P a bottle, or to sell it later at a
Find the exact area between y = x2 and y = x.
Find an antiderivative.ℎ(y) = 3y2 − y3
Find the integrals in Problem. Check your answers by differentiation. sin 0(cos 0 + 5)7 de
Urologists are physicians who specialize in the health of the bladder. In a common diagnostic test, urologists monitor the emptying of the bladder using a device that produces two graphs. In one of
Find the integrals in Problem. Check your answers by differentiation. [s sin° Ꮎ cos Ꮎ dᎾ
Figure 6.13 shows the derivative F' of F. Let F(0) = 0. Of the four numbers F(1), F(2), F(3), and F(4), which is largest? Which is smallest? How many of these numbers are negative? F'(x) 2 3 Figure
Find the integrals in Problems. Check your answers by differentiation. /v Vcos 3t sin 3t dt
Use the Fundamental Theorem to find the area between f(x) = 2x and g(x) = 6 − x from the vertical axis to their point of intersection.
Find the exact area.Under y = te−t for 0 ≤ t ≤ 2.
During photosynthesis, plants absorb sunlight and release oxygen. The rate at which a leaf releases oxygen, the rate of photosynthesis, is a function of the age of the leaf. Figure 6.14 shows the
Find an antiderivative.k(x) = 10 + 8x3
The birth rate, B, in births per hour, of a bacteria population is given in Figure 6.15. The curve marked D gives the death rate, in deaths per hour, of the same population.(a) Explain what the shape
Find the exact area of the region bounded by the x-axis and the graph of y = x3 − x.
Find an antiderivative.p(r) = 2πr
Use the Fundamental Theorem of Calculus to find the average value of f(x) = e0.5x between x = 0 and x = 3. Show the average value on a graph of f(x).
The concentration, C, in ng/ml, of a drug in the blood as a function of the time, t, in hours since the drug was administered is given by C = 15te−0.2t . The area under the concentration curve is a
Find the integrals in Problem. Check your answers by differentiation. [ sin(3-1) dt
Find an antiderivative.f(x) = x + x5 + x−5
Find the integrals in Problem. Check your answers by differentiation. ر 'x2x3+ +1 dx
Use Figure 6.17 and the fact that F(2) = 3 to sketch the graph of F(x). Label the values of at least four points. Area = 2 frie + F'(x) 12 3 4 5 6 7 Area = 7 Figure 6.17 Area = 4 +x 8

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