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mathematics
applied calculus

Questions and Answers of Applied Calculus

The marginal cost and marginal revenue of a company are MC(q) = 0.03q2 − 1.4q + 34 and MR(q) = 30, where q is the number of items manufactured. To increase profits, should the company increase or
Substitute t = 0, 10, 20,…, 70 into the exponential function used in this section to model the US population 1790–1860. Compare the predicted values of the population with the actual values.
Figure 4.102 shows a graph of the percentage of drug dissolved against time for four tetracycline products A, B, C, and D. Figure 4.103 shows the drug concentration curves for the same four
Graph a function with the given properties.Has local and global minimum at x = 3, local and global maximum at x = 8.
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
(a) Use the derivative to find all critical points.(b) Use a graph to classify each critical point as a local minimum, a local maximum, or neither.f(x) = x5 − 15x3
Figure 6.18 shows the derivative F'. If F(0) = 14, graph F. Give (x, y)-coordinates of all local maxima and minima. Area = 25 2 3 Area = 34 4 F'(x) Figure 6.18 5 6 Area = 5 X
If m, n ≥ 2 are integers, find and classify the critical points of f(x) = xm(1 − x)n.
Find the integrals in problem. Check your answers by differentiation. xe dx 3x tehra
Find the integrals in Problem. Check your answers by differentiation. [x√3x² + 4dx
For the equations (I) in Problem 8, what is the threshold value of S?Data from in problem 8 Compare the diseases modeled by each of the follow- ing differential equations with the flu model in this
For the equations (II) in Problem 8, suppose S0 = 100. Does the disease spread initially? What if S0 = 200?Data from in problem 8 Compare the diseases modeled by each of the follow- ing differential
Compare the diseases modeled by each of the following differential equations with the flu model in this section. Match each set of differential equations with one of the following statements. Write a
At a time t hours after taking a tablet, the rate at which a drug is being eliminated is r(t) = 50(e−0.1t − e−0.2t) mg/hr.Assuming that all the drug is eventually eliminated, calculate the
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope
Find the derivative. Assume a, b, c, k are constants.y = x−12
Indicate the approximate locations of all inflection points. How many inflection points are there? f(t) t
Figure 4.95 shows drug concentration curves for anhydrous ampicillin for newborn babies and adults. Discuss the differences between newborns and adults in the absorption of this drug. drug
Problem are about chikungunya, a disease that arrived in the Americas in 2013 and spread rapidly in 2014. While seldom fatal, the disease causes debilitating joint pain and a high fever. In August
Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. f(t) t
In Problems for f(x) in Figure 4.39, find the x-values of the global maximum and global minimum on the given domain.Domain: 0 ≤ x ≤ 4 60 40 20 -3 -2 -1 1 2 Figure 4.39 f(x) 3 4 X
The graph of a cost function is given in Figure 4.66.(a) At q = 25, estimate the following quantities and represent your answers graphically.(i) Average cost (ii) Marginal cost(b) At approximately
The cost of producing q items is C(q) = 2500 + 12q dollars.(a) What is the marginal cost of producing the 100th item? the 1000th item?(b) What is the average cost of producing 100 items? 1000 items?
The elasticity of bread is E = 0.35. If the price of bread increases from $1.75 to $2.00, does the demand for bread increase or decrease? Estimate the percent change in the quantity demanded.
Revenue is given by R(q) = 450q and cost is given by C(q) = 10,000 + 3q2. At what quantity is profit maximized? What is the total profit at this production level?
Problem are about chikungunya, a disease that arrived in the Americas in 2013 and spread rapidly in 2014. While seldom fatal, the disease causes debilitating joint pain and a high fever. In August
Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. f(x) X
Indicate the approximate locations of all inflection points. How many inflection points are there? f(x) X
For Problems indicate all critical points on the given graphs. Determine which correspond to local minima, local maxima, global minima, global maxima, or none of these. 4 2 y - X 24 4 6 6 8 10
The elasticity of a good is E = 2. What is the effect on the quantity demanded of:(a) A 3% price increase?(b) A 3% price decrease?
Figure 4.65 shows cost with q = 10,000 marked.(a) Find the average cost when the production level is 10,000 units and interpret it.(b) Represent your answer to part (a) graphically.(c) At
The revenue from selling q items is R(q) = 500q − q2, and the total cost is C(q) = 150+10q. Write a function that gives the total profit earned, and find the quantity which maximizes the profit.
For each cost function in Figure 4.64, is there a value of q at which average cost is minimized? If so, approximately where? Explain your answer.(a)(b) (cost) с q (quantity)
Let b = 1, and graph C = ate−bt using different values for a. Explain the effect of the parameter a.
Figure 4.51 shows cost and revenue. For what production levels is the profit function positive? Negative? Estimate the production at which profit is maximized. $ (thousands) 400
The elasticity of a good is E = 0.5. What is the effect on the quantity demanded of:(a) A 3% price increase? (b) A 3% price decrease?
Indicate the approximate locations of all inflection points. How many inflection points are there? f(x)
Investigate the effect of the parameter C on the logistic curveSubstitute several values for C and explain, with a graph and with words, the effect of C on the graph. P = 10 1+ Ce-t
Indicate all critical points on the given graphs. Determine which correspond to local minima, local maxima, global minima, global maxima, or none of these. 8 4 y 1 2 3 4 5 x
Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. X (x) f
If time, t, is in hours and concentration, C, is in ng/ml, the drug concentration curve for a drug is given by C = 12.4te−0.2t.(a) Graph this curve.(b) How many hours does it take for the drug
The cost (in dollars) of producing q items is given byC(q) = 0.08q3 + 75q + 1000.(a) Find the marginal cost function.(b) Find C(50) and C¨(50). Give units with your answers and explain what each is
The demand for a product is given, for p, q ≥ 0, byp = f(q) = 50 − 0.03q2.(a) Find the p- and q-intercepts for this function and interpret them in terms of demand for this product.(b) Find f(20)
The yield, Y , of an apple orchard (measured in bushels of apples per acre) is a function of the amount x of fertilizer in pounds used per acre. SupposeY = f(x) = 320 + 140x − 10x2.(a) What is the
The demand curve for a product is given by q = 300 − 3p, where p is the price of the product and q is the quantity that consumers buy at this price.(a) Write the revenue as a function, R(p), of
A ball is dropped from the top of the Empire State Building. The height, y, of the ball above the ground (in feet) is given as a function of time, t (in seconds), byy = 1250 − 16t2.(a) Find the
The cost to produce q items is C(q) = 1000 + 2q2 dollars. Find the marginal cost of producing the 25th item. Interpret your answer in terms of costs.
Suppose W is proportional to r3. The derivative dW∕dr is proportional to what power of r?
(a) Use the formula for the area of a circle of radius r, A = πr2, to find dA∕dr.(b) The result from part (a) should look familiar. What does dA∕dr represent geometrically?(c) Use the difference
An island of A square kilometers in the Florida Keys has N species of beetles, given byN = f(A) = 8.3A0.25.(a) Find f¨(A).(b) At 40 km2, Key Largo is the largest island in the Keys. Find f(40) and
A jökulhlaup is the rapid draining of a glacial lake when an ice dam bursts. The maximum outflow rate, Q (in m3∕sec), during a jökulhlaup is given1 in terms of its volume, v (in km3), before the
If you are outdoors, the wind may make it feel a lot colder than the thermometer reads. You feel the windchill temperature, which, if the air temperature is 20◦F, is given in ◦F by W (v) = 48.17
Circulation time is the average amount of time it takes for all the blood in the body to circulate once and return to the heart. For a mammal with mass m kilograms, the circulation time C, in
Kleiber’s Law states that the daily calorie requirement, C(w), of a mammal is proportional to the mammal’s body weight w raised to the 0.75 power. If body weight is measured in pounds, the
The time, T, in seconds for one complete oscillation of a pendulum is given by T = f(L) = 1.111 √L, where L is the length of the pendulum in feet. Find the following quantities, with units, and
For a certain type of fish, the weight y, in grams, is related to the length x, in cm, by y = f(x) = 0.009x3.(a) Find f'(x).(b) Assume f(A) = B and f'(C) = D. For each of A, B, C, and D, which of the
(a) Find the equation of the tangent line to f(x) = x3 at the point where x = 2.(b) Graph the tangent line and the function on the same axes. If the tangent line is used to estimate values of
Find the quadratic polynomial g(x) = ax2 + bx + c which best fits the function f(x) = ex at x = 0, in the sense thatg(0) = f(0), and g'(0) = f'(0), and g''(0) = f''(0).Using a computer or calculator,
Find the equation of the line tangent to the graph of f(t) = 6t − t2 at t = 4. Sketch the graph of f(t) and the tangent line on the same axes.
Food bank usage in Britain has grown dramatically over the past decade. The number of users, in thousands, of the largest bank is estimated to be N(t) = 1.3(2.25)t , where t is the number of years
For f(x) = x2 + 5x +2, find the equation of the tangent line at(a) x = 1 (b) x = −1
(a) Use Figure 3.11 to rank the quantities f¨(−1), f¨(0), f¨(1), f¨(4) from smallest to largest.(b) Confirm your answer by calculating the quantities using the formula, f(x) = x3 − 3x2 + 2x +
For the functions in Problems(a) Find the derivative at x = −1.(b) Find the second derivative at x = −1.(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in
In 2009, the population, P, of India was 1.166 billion and growing at 1.5% annually.(a) Give a formula for P in terms of time, t, measured in years since 2009.(b)What do each of these represent in
(a) Differentiating, we have k'(x) = 3x2 − 1, so For the functions in Problems (a) Find the derivative at x = −1.(b) Find the second derivative at x = −1.(c) Use your answers to parts (a) and
Find the equation of the line tangent to the graph of f(x) = 2x3 − 5x2 + 3x − 5 at x = 1.
For the functions in Problems(a) Find the derivative at x = −1.(b) Find the second derivative at x = −1.(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in
With t in years since the start of 2014, worldwide annual extraction of copper is 17.9(1.025)t million tonnes.(a) How fast is the annual extraction changing at time t? Give units.(b) How fast is the
For the functions in Problems(a) Find the derivative at x = −1.(b) Find the second derivative at x = −1.(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in
(a) Find the equation of the tangent line to y = ln x at x = 1.(b) Use it to calculate approximate values for ln(1.1) and ln(2).(c) Using a graph, explain whether the approximate values are smaller
If a someone is lost in the wilderness, the search and rescue team identifies the boundaries of the search area and then uses probabilities to help optimize the chances of finding the person,
In 2012, the population of Mexico was 115 million and growing 1.09% annually, while the population of the US was 314 million and growing 0.9% annually.(a) Find the Mexican growth rate in people/year
For positive constants c and k, the Monod growth curve describes the growth of a population, P, as a function of the available quantity of a resource, r:Find dP∕dr and interpret it in terms of the
For the cost function C = 1000 + 300 ln q (in dollars), find the cost and the marginal cost at a production level of 500. Interpret your answers in economic terms.
Carbon-14 is a radioactive isotope used to date objects. If A0 represents the initial amount of carbon-14 in the object, then the quantity remaining at time t, in years, isA(t) =
Show that if the graphs of f(t) and ℎ(t) = Aekt are tangent at t = a, then k is the relative rate of change of f at t = a.
If ℎ = fn, show that (f") if = n² f
Find the value of c in Figure 3.19, where the line l tangent to the graph of y = 2x at (0, 1) intersects the x-axis. С y Figure 3.19 y = 2x X
If f(t) = t4 − 3t2 + 5t, find f'(t) and f''(t).
The cost of producing a quantity, q, of a product is given byC(q) = 1000 + 30e0.05q dollars.Find the cost and the marginal cost when q = 50. Interpret these answers in economic terms.
Some economists suggest that an extra year of education increases a person’s wages, on average, by about 14%. Assume you could make $10 per hour with your current level of education and that
If f(t) = 2t3 − 4t2 + 3t − 1, find f'(t) and f''(t).
Use Figure 3.22 to evaluate the derivative. 80 0 # f(x) 80 $6 0 Figure 3.22 80
At a time t hours after it was administered, the concentration of a drug in the body is f(t) = 27e−0.14t ng/ml. What is the concentration 4 hours after it was administered? At what rate is the
Given y = f(x) with f(1) = 4 and f'(1) = 3, find(a) g'(1) if g(x) = √f(x).(b) ℎ'(1) if ℎ(x) = f(√x).
The quantity, Q, in tons, of material at a municipal waste site is a function of the number of years since 2000, withQ = f(t) = 3t2 + 100.Find f(10), f¨(10), and the relative rate of change f¨∕f
Use Figure 3.22 to evaluate the derivative. 80 0 # f(x) 80 $6 0 Figure 3.22 80
Use Figure 3.22 to evaluate the derivative. 80 0 # f(x) 80 $6 0 Figure 3.22 80
Zebra mussels are freshwater shellfish that first appeared in the St. Lawrence River in the early 1980s and have spread throughout the Great Lakes. Suppose that t months after they appeared in a
Show that the relative rate of change of a quotient f∕g is the difference between the relative rates of change of f and g.
With a yearly inflation rate of 2%, prices are given byP = P0(1.02)t,where P0 is the price in dollars when t = 0 and t is time in years. Suppose P0 = 1. How fast (in cents/year) are prices rising
If C (in units of 10−4 molar) is the concentration of glucose in a solution, then E. coli bacteria in the solution grow at a rate, R (in cell divisions per hour), given by(a) Find the growth rate
The number, N, of acres of harvested land in a region is given byN = f(t) = 120√t,where t is the number of years since farming began in the region. Find f(9), f'(9), and the relative rate of change
Use Figure 3.22 to evaluate the derivative. 80 0 # f(x) 80 $6 0 Figure 3.22 80
Show that the relative rate of change of a product fg is the sum of the relative rates of change of f and g.
With t in years since January 1, 2010, the population P of Slim Chance is predicted byP = 35,000(0.98)t.At what rate will the population be changing on January 1, 2023?
Use Figure 3.22 to evaluate the derivative. 80 0 # f(x) 80 $6 0 Figure 3.22 80
Use the tangent line approximation.Given f(x) = x3 + x2 − 6, approximate f(0.97).
The quantity, q, of a skateboard sold depends on the selling price, p, in dollars, so we write q = f(p). You are given that f(140) = 15,000 and f¨(140) = −100.(a) What do f(140) = 15,000 and
The population of Hungary has been modeled byP = 9.919(0.998)t,where P is in millions and t is years since July 2014.(a) What does this model predict for the population of Hungary in July 2020?(b)

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