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mathematics
applied calculus
Applied Calculus 6th Edition Deborah Hughes Hallett, Patti Frazer Lock, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale - Solutions
Use the Fundamental Theorem to evaluate the definite integral exactly. -3 [e f dt
Show analytically that if elasticity of demand satisfies E > 1, then the derivative of revenue with respect to price satisfies dR∕dp < 0.
Find the value of x that maximizes y = 12 + 18x −5x2 and the corresponding value of y, by(a) Estimating the values from a graph of y.(b) Finding the values using calculus.
Figure 4.27 shows the graph of the derivative, f'.(a) Which of the x-values A, B, C, D, E, F, and G appear to be critical points of f?(b) Which appear to be inflection points of f?(c) How many local maxima does f appear to have? How many local minima? A B C D E F G 1 1 Figure 4.27 X
The continuous function has a critical point.(a) Is the critical point a local maximum or a local mini-mum?(b) Sketch the graph near the critical point. Label the coordinates of the critical point.f(1) = 5, f'(1) = 0, f''(1) = −2
An online seller of knitted sweaters finds that it costs $35 to make her first sweater. Her cost for each additional sweater goes down until it reaches $25 for her 100th sweater, and after that it starts to rise again. If she can sell each sweater for $35, is the quantity sold that maximizes her
Show analytically that if elasticity of demand satisfies E < 1, then the derivative of revenue with respect to price satisfies dR∕dp > 0.
The continuous function has a critical point.(a) Is the critical point a local maximum or a local mini-mum?(b) Sketch the graph near the critical point. Label the coordinates of the critical point.g(−5) = 4, g'(−5) = 0, g''(−5) = 2
Sketch a possible graph of y = f(x), using the given information about the derivatives y' = f' (x) and y'' = f''(x). Assume that the function is defined and continuous for all real x. y
The continuous function has exactly one critical point. Find the x-values at which the global maximum and the global minimum occur in the interval given.f'(1) = 0, f''(1) = −2 on 1 ≤ x ≤ 3
The continuous function has a critical point.(a) Is the critical point a local maximum or a local mini-mum?(b) Sketch the graph near the critical point. Label the coordinates of the critical point.ℎ(2) = −5, ℎ'(2) = 0, ℎ''(2) = −4
A warehouse selling cement has to decide how often and in what quantities to reorder. It is cheaper, on average, to place large orders, because this reduces the ordering cost per unit. On the other hand, larger orders mean higher storage costs. The warehouse always reorders cement in the same
Dwell time, t, is the time in minutes that shoppers spend in a store. Sales, s, is the number of dollars they spend in the store. The elasticity of sales with respect to dwell time is 1.3. Explain what this means in simple language.
The continuous function has exactly one critical point. Find the x-values at which the global maximum and the global minimum occur in the interval given.g'(−5) = 0, g''(−5) = 2 on −6 ≤ x ≤ −5
The continuous function has a critical point.(a) Is the critical point a local maximum or a local mini-mum?(b) Sketch the graph near the critical point. Label the coordinates of the critical point.j(3) = 5, j'(3) is undefined, j'(x) = −1 for x < 3 and j'(x) = 1 for x > 3.
(a) Figure 4.27 shows the graph of f. Which of the x- values A,B, C,D,E, F, and G appear to be critical points of f?(b) Which appear to be inflection points of f?(c) How many local maxima does f appear to have? How many local minima? A B C D E F G 1 1 Figure 4.27 X
Find all critical points and then use the first-derivative test to determine local maxima and minima. Check your answer by graphing. f(x) = x x² +1
A linear demand function is given in Figure 4.68. Economists compute elasticity of demand E for any quantity q0 using the formulaE = d1∕d2,where d1 and d2 are the vertical distances shown in Figure 4.68.(a) Explain why this formula works.(b) Determine the prices, p, at which(i) E > 1 (ii) E
For the functions in Problems do the following:(a) Find f' and f'' .(b) Find the critical points of f.(c) Find any inflection points of f.(d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval.(e) Graph
Consider a population P satisfying the logistic equation(a) Use the chain rule to find d2P∕dt2.(b) Show that the point of diminishing returns, where d2P∕dt2 = 0, occurs where P = L∕2. dP = kp (1-2). dt
Figure 4.26 is the graph of a second derivative f''. On the graph, mark the x-values that are inflection points of f. f" (x) AAY Figure 4.26 X
Suppose cost is proportional to quantity, C(q) = kq. Show that a firm earns maximum profit when Profit 1 Revenue E
Cell membranes contain ion channels. The fraction, f, of channels that are open is a function of the membrane potential V (the voltage inside the cell minus voltage outside), in millivolts (mV), given by(a) Find the values of L, k, and C in the logistic formula for f:(b) At what voltages V are 10%,
The demand equation for a quantity q of a product at price p, in dollars, is p = −5q + 4000. Companies producing the product report the cost, C, in dollars, to produce a quantity q is C = 6q + 5 dollars.(a) Express a company’s profit, in dollars, as a function of q.(b) What production level
Find all critical points and then use the first-derivative test to determine local maxima and minima. Check your answer by graphing.f(x) = (x3 − 8)4
For the functions in Problems do the following:(a) Find f' and f''.(b) Find the critical points of f.(c) Find any inflection points of f.(d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval.(e) Graph
A farmer uses x lb of fertilizer per acre at a cost of $2 per pound, leading to a revenue of R = 700−400e−x∕100 dollars per acre.(a) How many pounds of fertilizer should be applied per acre to maximize profit?(b) What is the maximum profit on a 200 acre farm?
(a) Find all critical points and all inflection points of the function f(x) = x4 − 2ax2 + b. Assume a and b are positive constants.(b) Find values of the parameters a and b if f has a critical point at the point (2, 5).(c) If there is a critical point at (2, 5), where are the inflection points?
If p is price and E is the elasticity of demand for a good, show analytically that Marginal revenue = p(1 − 1∕E).
Find all critical points and then use the first-derivative test to determine local maxima and minima. Check your answer by graphing.f(x) = (x2 − 4)7
At a price of $8 per ticket, a musical theater group can fill every seat in the theater, which has a capacity of 1500. For every additional dollar charged, the number of people buying tickets decreases by 75. What ticket price maximizes revenue?
A population, P, growing logistically is given by(a) Show that(b) Explain why part (a) shows that the ratio of the additional population the environment can support to the existing population decays exponentially. P = L 1+ Ce-kt
(a) Use a graph to estimate the x-values of any critical points and inflection points of f(x) = e−x2.(b) Use derivatives to find the x-values of any critical points and inflection points exactly.
For Problems sketch a possible graph of y = f(x), using the given information about the derivatives y' = f'(x) and y'' = f''(x). Assume that the function is defined and continuous for all real x. y' = 0 ! _y' > 0 | X1 1 y" = 0 y V' 0 x2 1 y' = 0 >0 y = 0 | y < 0 + x3 y"
A landscape architect plans to enclose a 3000-squarefoot rectangular region in a botanical garden. She will use shrubs costing $45 per foot along three sides and fencing costing $20 per foot along the fourth side. Find the minimum total cost.
You run a small furniture business. You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. The price will be $90 per chair up to 300 chairs, and above 300, the price will be reduced by $0.25 per chair (on the whole order) for every
If q is the quantity of chicken demanded as a function of the price p of beef, the cross-price elasticity of demand for chicken with respect to the price of beef is defined as Ecross = |p∕q ⋅ dq∕dp|. What does Ecross tell you about the sensitivity of the quantity of chicken bought to changes
Find all critical points and then use the first-derivative test to determine local maxima and minima. Check your answer by graphing.f(x) = 3x4 − 4x3 + 6
Show that a demand equation q = k∕pr, where r is a positive constant, gives constant elasticity E = r.
An ice cream company finds that at a price of $4.00, demand is 4000 units. For every $0.25 decrease in price, demand increases by 200 units. Find the price and quantity sold that maximize revenue.
There are two kinds of dose-response curves. One type, discussed in this section, plots the intensity of response against the dose of the drug. We now consider a dose-response curve in which the percentage of subjects showing a specific response is plotted against the dose of the drug. In Problems
For the functions in Problems do the following:(a) Find f' and f'' .(b) Find the critical points of f.(c) Find any inflection points of f.(d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval.(e) Graph
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither.f(x) = 3x5 − 5x3
Figure 4.41 shows the rate at which photosynthesis is taking place in a leaf.(a) At what time, approximately, is photosynthesis proceeding fastest for t ≥ 0?(b) If the leaf grows at a rate proportional to the rate of photosynthesis, for what part of the interval 0 ≤ t ≤ 200 is the leaf
Find the critical points of the function and classify them as local maxima or minima or neither.ℎ(x) = x + 1∕x
(a) If the demand equation is pq = k for a positive constant k, compute the elasticity of demand.(b) Explain the answer to part (a) in terms of the revenue function
The demand for tickets to an amusement park is given by p = 70 − 0.02q, where p is the price of a ticket in dollars and q is the number of people attending at that price.(a) What price generates an attendance of 3000 people? What is the total revenue at that price? What is the total revenue if
There are two kinds of dose-response curves. One type, discussed in this section, plots the intensity of response against the dose of the drug. We now consider a dose-response curve in which the percentage of subjects showing a specific response is plotted against the dose of the drug. In Problems
Find the critical points of the function and classify them as local maxima or minima or neither.g(x) = xe−3x
(a) Let p be the price and q be the quantity sold of a good with a high elasticity of demand, E. Explain intuitively (without formulas) the effect of raising the price on the revenue, R.(b) Derive an expression for dR∕dp in terms of q and E. Show all the steps and reasoning.(c) Explain how your
The demand equation for a product is p = 45 − 0.01q. Write the revenue as a function of q and find the quantity that maximizes revenue. What price corresponds to this quantity? What is the total revenue at this price?
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither.f(x) = x4 − 4x3 + 10
Explain why it is safer to use a drug for which the derivative of the dose-response curve is smaller.
Graph the function and describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.f(x) = x + 2 sin x
Sketch the graph of a function on the interval 0 ≤ x ≤ 10 with the given properties.Has local and global minimum at x = 3, local and global maximum at x = 8.
Differentiate the functions in Problems. Assume that A, B, and C are constants.P = 200e0.12t
Find the derivative of the functions in Problems.f(t) = ln(t2 + 1)
Find the derivative. Assume a, b, c, k are constants.y = −3x4 − 4x3 − 6x
Find the derivative. Assume that a, b, c, and k are constants.f(z) = √ze−z
Differentiate the functions in Problems. Assume that A and B are constants.z = cos(4θ)
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = e−4t
Find the derivative of the functions in Problems.w = e√s
Find the derivative. Assume that a, b, c, and k are constants. f(x) = x² + 3 x
Find the derivative. Assume a, b, c, k are constants.y = 3x2 + 7x − 9
Differentiate the functions in Problems. Assume that A and B are constants.y = 6 sin(2t) + cos(4t)
Differentiate the functions in Problems. Assume that A, B, and C are constants.P = e−0.2t
Find the derivative of the functions in Problems.y = ln(5t + 1)
Find the derivative. Assume a, b, c, k are constants.y = 8t3 − 4t2 + 12t − 3
Find the derivative. Assume that a, b, c, and k are constants.P = t2 ln t
Differentiate the functions in Problems. Assume that A and B are constants.y = sin(x2)
Differentiate the functions in Problems. Assume that A, B, and C are constants.f(t) = e3t
Find the derivative of the functions in Problems.y = 5e5t+1
Use the definition of the derivative to show that if f(x) = u(x) + v(x), for functions u(x) and v(x), then f'(x) = u'(x) + v'(x).
Find the derivative. Assume a, b, c, k are constants.y = x2 + 5x + 9
For Problems find the derivative. Assume that a, b, c, and k are constants.R = 3qe−q
Differentiate the functions in Problems. Assume that A and B are constants.y = Asin(Bt)
Find the derivative of the functions in Problems.< w = e−3t2
Use the definition of the derivative to show that if f(x) = k ⋅ u(x), where k is a constant and u(x) is a function, then f'(x) = k ⋅ u'(x).
For Problems find the derivative. Assume a, b, c, k are constants.y = 6x3 + 4x2 − 2x
Find the derivative. Assume that a, b, c, and k are constants.z = (3t + 1)(5t + 2)
Differentiate the functions in Problems. Assume that A and B are constants.W = 4 cos(t2)
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = 3x − 2 ⋅ 4x
Find the derivative of the functions in Problems.C = 12(3q2 − 5)3
For Problems find the derivative. Assume a, b, c, k are constants.f(x) = 1/x4
For Problems find the derivative. Assume that a, b, c, and k are constants.y = (t3 − 7t2 + 1)et
Differentiate the functions in Problems. Assume that A and B are constants.y = 2 cos(5t)
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = 5 ⋅ 2x − 5x + 4
Find the derivative of the functions in Problems.f(x) = 6e5x + e−x2
Find the derivative. Assume a, b, c, k are constants.f(q) = q3 + 10
(a) Use a graph of g(ℎ) =2ℎ − 1/ℎ to explain why we believe that limℎ→0 2ℎ − 1/ℎ ≈ 0.6931.(b) Use the definition of the derivative and the result from part (a) to explain why, if f(x) = 2x, we believe that f'(x) ≈ (0.6931)2x.
Find the derivative. Assume that a, b, c, and k are constants.y = (t2 + 3)et
Differentiate the functions in Problems. Assume that A and B are constants.f(x) = sin(3x)
Differentiate the functions in Problems. Assume that A, B, and C are constants.f(x) = 2x + 2 ⋅ 3x
Find the derivative of the functions in Problems.y = √s3 + 1
Find the derivative. Assume a, b, c, k are constants.y = 3t4 − 2t2
For Problems use the definition of the derivative to obtain the following results.If f(x) = x5, then f'(x) = 5x4.
For Problems find the derivative. Assume that a, b, c, and k are constants.y = x ln x
Differentiate the functions in Problems. Assume that A and B are constants.R = sin(5t)
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = 4 ⋅ 10x − x3
Find the derivative of the functions in Problems.y = 12 − 3x2 + 2e3x
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