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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
At what points on the graph of ƒ(x) = 2x3 - 9x2 - 12x + 5 is the slope of the tangent line 12?
Consider the following table of values of the functions f and g and their derivatives at various points.Find the following using the table above.(a) Dx (ƒ [ g(x)]) at x = 1(b) Dx 1ƒ [ g(x2]) at x = 2 X f(x) f'(x) g(x) g'(x) 1 2 -6 2 2/7 2 4 -7 3 3/7 3 1 -8 4 4/7 4 3 -9 1 5/7
Explain why the general power rule is a consequence of the chain rule and the power rule.
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms.ƒ(x) = ln√5x - 3
Consider the following table of values of the functions f and g and their derivatives at various points.Find the following using the table.(a) Dx (ƒ [ g(x)]) at x = 2 (b) Dx (ƒ [ g(x)]) at x = 3 1 2 X f(x) 3 4 f'(x) -5 -6 g(x) 4 1 g'(x) 2/9 3/10 3 2 -7 2 4/11 4 1 -11 3 6/13
Use the rules for derivatives to find the derivative of each function defined as follows. f(x) = = In(√x + 1)
Use the rules for derivatives to find the derivative of each function defined as follows.ƒ(x) = e2x ln(xex + 1)
The total cost (in dollars) to produce x T-shirts is C(x) = 0.02x2 + 300. Find and interpret the following.(a) C(50) (b) C′(50)(c) C̅(50) (d) C̅′(50)
The total cost (in hundreds of dollars) to produce x units of perfume is(a) Find the average cost function.(b) Find the marginal average cost function.(c) Find the average cost and the marginal average cost for a production level of 10 units. Interpret your results.(d) Find the average cost and the
At what points on the graph of ƒ(x) = x3 + 6x2 + 21x + 2 is the slope of the tangent line 9?
Consider the following table of values of the functions f and g and their derivatives at various points.Find the following using the table.(a) Dx (g [ƒ(x)]) at x = 2 (b) Dx (g [ ƒ(x)]) at x = 3 1 2 X f(x) 3 4 f'(x) -5 -6 g(x) 4 1 g'(x) 2/9 3/10 3 2 -7 2 4/11 4 1 -11 3 6/13
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms.g(x) = ln√3x2 - 7
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line. y 2 x² X ; x = 2
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line. y 3 x - l' x = -1
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = x2 - 6x; x = 2
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = 8 - x2; x = 1
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = √6x - 2; x = 3
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = -√8x + 1; x = 3
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = ex; x = 0
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = xex; x = 1
In Exercise, use the ideas from Exercise 67 to find the derivative of each unction.h(x) = xxData from Exercise 67Let h(x) = u(x)v(x).Using the fact that use the chain rule, the product rule, and the formula for the derivative of ln x to show thatUse the result from part (a) and the fact
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = ln x; x = 1
Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line.y = x ln x; x = e
Suppose that the student body in your college grows by 2% and the tuition goes up by 3%. Use the result from the previous exercise to calculate the approximate amount that the total tuition collected goes up, and compare this with the actual amount.
Why is e a convenient base for exponential and logarithmic functions?
In Exercise 48 in Section 4.2, we saw an example for which the erroneous version of the product rule, [ƒ(x)g(x)]′= ƒ′(x)g′(x), does hold. As a more general version of this example, calculate [ƒ(x)g(x)]′and ƒ′(x)g′(x) for ƒ(x) = a/(n - x)n and g(x) = bxn for n ≠ 0, and verify that
Just as there are times when the erroneous version of the product rule holds there are times when the erroneous version of the quotient rule holds. Calculate [ƒ(x)g(x)]′ and ƒ′(x)g′(x) for ƒ(x) = aek2x/(k-1) and g(x) = bekx for k ≠ 1, and verify that they are equal.
If a sum of $1000 is deposited into an account that pays r % interest compounded quarterly, the balance after 12 years is given by Find and interpret dA/dr when r = 5. A 1000 1 + = 1000 (1 400 48
Find the marginal average cost function of each function defined as follows.C(x) = √x + 1
Suppose that the profit (in hundreds of dollars) from selling x units of a product is given byFind and interpret the marginal profit when the following numbers of units are sold.(a) 4(b) 12(c) 20(d) What is happening to the marginal profit as the number sold increases?(e) Find and interpret the
A company finds that its total costs are related to the amount spent on training programs bywhere T(x) is costs in thousands of dollars when x hundred dollars are spent on training. Find and interpret T′(x) if the following amounts are spent on training.(a) $900(b) $1900(c) Are costs per dollar
After declining over the last century, the percentage of men aged 65 and older in the workforce has begun to rise in recent years, as shown by the following table. (a) Using the regression feature on a graphing calculator, find a cubic function that models this data, letting t = 0 correspond
Find the marginal average cost function of each function defined as follows.C(x) = 10 - e-x
If a sum of money is deposited into an account that pays r% interest compounded annually, the doubling time (in years) is given byFind and interpret dT/dr when r = 5. T = In 2 In(1 + r/100)
Find the marginal average cost function of each function defined as follows.C(x) = ln(x + 5)
The sales of a company are related to its expenditures on research by S(x) = 1000 + 60√x + 12x, where S(x) gives sales in millions when x thousand dollars is spent on research. Find and interpret dS/dx if the following amounts are spent on research.(a) $9000(b) $16,000(c) $25,000(d) As the amount
The U.S. dollar has been declining in value over the last century, except during the Great Depression, when it increased in value. The following table shows the number of dollars it took in various years to equal $1 in 1913.(a) Using the regression feature on a graphing calculator, find a cubic
If a sum of $1000 is deposited into an account that pays r% interest compounded continuously, the balance after 12 years is given by A = 1000e12r/100.Find and interpret dA/dr when r = 5.
The distance from Belmar the cat to a piece of string he is stalking is given in feet bywhere t is the time in seconds since he begins.(a) Find Belmar’s average velocity between 1 second and 3 seconds.(b) Find Belmar’s instantaneous velocity at 3 seconds. f(t) = 8 20 + +2 t + 1 f² + 1'
The number (in billions) of pieces of mail handled by the U.S. Post Office each year from 2010 through 2019 can be approximated by P(t) = -0.003584t4 - 0.01051t3 + 0.7432t2 - 6.508t + 171.7 where t is the number of years since 2010. Find and interpret the rate of change in the volume of mail for
Suppose a population is growing exponentially with an annual growth constant k = 0.05. How fast is the population growing when it is 1,000,000? Use the derivative to calculate your answer, and then explain how the answer can be obtained without using the derivative.
Suppose a population is growing logistically with k = 5 * 10-6, m = 30,000, and G0 = 2000. Assume time is measured in years.(a) Find the growth function G(t) for this population.(b) Find the population and rate of growth of the population after 6 years. Interpret your results.
The length of the monkeyface prickleback, a West Coast game fish, can be approximated by L = 71.5(1 - e-0.1t) and the weight by W = 0.01289 • L2.9, where L is the length in centimeters, t is the age in years, and W is the weight in grams.(a) Find the approximate length of a 5-year-old
The age/weight relationship of male Arctic foxes caught in Svalbard, Norway, can be estimated by the function M(t) = 3583e-e-0.020(t-66), where t is the age of the fox in days and M(t) is the weight of the fox in grams.(a) Estimate the weight of a male fox that is 250 days old.(b) Use M(t) to
We found that if a rat initially has 10,000 Salmonella bacteria, then the projected number of bacteria in t hours can be approximated by y(t) = 10,000(1.149)t.Find and interpret the instantaneous rate of change in the projected number of bacteria at each of the following times.(a) After 2
We found that the total world wind energy capacity (in megawatts) in recent years could be approximated by the function C(t) = 49,330.3(1.14853)t, where t is the number of years since 2000. Find the rate of change in the energy capacity for the following years.(a) 2010 (b) 2015 (c) 2020
We found that the production of corn (in billions of bushels) in the United States since 1930 could be approximated by p(t) = 1.757(1.0248)t, where t is the number of years since 1930. Find and interpret p′(2020).
Over time, the number of original basic words in a language tends to decrease as words become obsolete or are replaced with new words. Linguists have used calculus to study this phenomenon and have developed a methodology for dating a language, called glottochronology. Experiments have indicated
A study by the National Highway Traffic Safety Administration found that driver fatalities rates were highest for the youngest and oldest drivers. The rates per 1000 licensed drivers for every 100 million miles may be approximated by the function ƒ(x) = k(x - 49)6 + 0.8, where x is the driver’s
In Exercises, find f[g(x) ] and g[f(x)]. f(x) = 2 √4; g(x) = 2 − x x4
Use the rules for derivatives to find the derivative of each function defined as follows.y = -4x-3
In Exercises, use the product rule to find the derivative of each function.q(x) = (x-2 - x-3)(3x-1 + 4x-4)
In Exercises, use the quotient rule to find the derivative of each function. y = 5 - 3t 4+ t
In Exercises, find f[g(x)] and g[f(x)]. f(x) = x² + 1; g(x) = e*-1
In Exercises, use the quotient rule to find the derivative of each function. y = x² + x x - 1
Find the derivative of each function. y = 2 ln(x + 3) x² 2
In Exercises, find the derivative of each function. y || 6 x4 7 .3 X 3 + + √5 X
Use the rules for derivatives to find the derivative of each function defined as follows. y = 2x³ - 5x² x + 2
In Exercises, find f[g(x) ] and g[f(x)]. f(x) = X g(x) = x²
In Exercises, use the quotient rule to find the derivative of each function. f(x) = 6х + 1 3x + 10
In Exercises, use the product rule to find the derivative of each function.p(y) = (y-1 + y-2)(2y-3 - 5y-4)
In Exercises, find f[g(x)] and g[f(x)]. ) = √x + 2; g(x) = 8x² - 6 f(x)
In Exercises, find the derivative of each function.y = 8√x + 6x3/4
Find the derivative of each function.y = ln(x4 + 5x2)3/2
In Exercises, use the quotient rule to find the derivative of each function. f(x) = 8x11 7x + 3
Find the derivative of each function.y = ln(5x3 - 2x)3/2
Use the rules for derivatives to find the derivative of each function defined as follows. k(x) = 3x 4x + 7
Use the rules for derivatives to find the derivative of each function defined as follows.ƒ(x) = 3x-4 + 6√x
In Exercises, find f[g(x) ] and g[f(x)]. f(x) = 9x² 11x; g(x) = 2√x + 2
In Exercises, find the derivative of each function. f(t) = 7 5 1³
In Exercises, find the derivative of each function.y = 10x-3 + 5x-4 - 8x
Find the derivative of each function.y = -5x ln(3x + 2)
Use the rules for derivatives to find the derivative of each function defined as follows. r(x) = -8x 2x + 1
Use the rules for derivatives to find the derivative of each function defined as follows.ƒ(x) = 19x-1 - 8√x
In Exercises, find the derivative of each function.y = 5x-5 - 6x-2 + 13x-1
Find the derivative of each function.y = (3x + 7) ln(2x - 1)
In Exercises, use the quotient rule to find the derivative of each function. y 9 - 7t 1 - t
In Exercises, find f[g(x) ] and g[f(x)]. f(x)= In(x² + 4); g(x) = 2x - 1
In Exercises, find the derivative of each function. f(t) 14 = + t 12 + √₂ V2
Use the rules for derivatives to find the derivative of each function defined as follows. x - 1 I + x - zx y = ||
Find the derivative of each function.s = t2 ln|t|
Find the derivative of each function.y = x ln |2 - x2|
In Exercises, use the quotient rule to find the derivative of each function. y x² - 4x x + 3
Write each function as the composition of two functions. (There may be more than one way to do this.)y = (5 - x2)3/5
In Exercises, find the derivative of each function. y 3 .6 X + 1 .5 X 7 2 X
Find the derivative of each function. v = In u u³
In Exercises, use the quotient rule to find the derivative of each function. f(t) = 41² + 11 1² + 3
Write each function as the composition of two functions. (There may be more than one way to do this.)y = (3x2 - 7)2/3
Use the rules for derivatives to find the derivative of each function defined as follows.ƒ(x) = (3x2 - 2)4
Determine whether each statement is true or false, and explain why. The derivative of ƒ(x) = 1/x4 is ƒ′(x) = 1/(4x3).
Determine whether each statement is true or false, and explain why.The derivative of g(x) = log |x| is the same as the derivative of h(x) = log x.
The marginal average cost function estimates how the cost per item is changing at a production of x units.Determine whether each statement is true or false, and explain why.
Find the derivative of each function.y = ln(8x)
Find the derivative of each function. y In x 4x + 7
In Exercises, use the quotient rule to find the derivative of each function. y || x8 + zx- 4x² - 5
Write each function as the composition of two functions. (There may be more than one way to do this.)y = -√13 + 7x
In Exercises, find the derivative of each function.p(x) = -10x-1/2 + 8x-3/2
Find the derivative of each function. У || = -2 In x 3x - 1
Write each function as the composition of two functions. (There may be more than one way to do this.)y = √9 - 4x
In Exercises, find the derivative of each function.h(x) = x-1/2 - 14x-3/2
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