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mathematics
applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

In Exercises , refer to a line of slope m. If you begin at a point on the line and move h units in the x-direction, how many units must you move in the y-direction to return to the line?m = 2/3 , h = 1/2
In Exercises, apply the three-step method to compute f′(x) for the given function. Follow the steps that we used in Example 6. Make sure to simplify the difference quotient as much as possible before taking limits.Example 6Computing a Derivative from the Limit Definition Use limits to compute the
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.f (x) = √x, x = 1/9Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.(x) = 1/x , x = .01Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the
If s = 7x2y√z, find:(a) d2s/dx2(b) d2s/dy2(c) ds/dz
Differentiate.g(P) = 4P0.7
In Exercises, we specify a line by giving the slope and one point on the line. We give the first coordinate of some points on the line. Without deriving the equation of the line, find the second coordinate of each point.Slope is 2, (1, 3) on line; (2, ); (3, ); (0, ).
In Exercises, use limits to compute f′(x).f (x) = 3x + 1
In Fig. 17, h represents a positive number, and 3 + h is the number h units to the right of 3. Draw line segments on the graph having the following lengths.(a) f(3) (b) f(3 + h)(c) f(3 + h) - f(3) (d) h(e) Draw a line of slope f(3+h)-f(3) h
Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. Interpret C(50) = 5000 and C′(50) = 45.
In Exercises, we specify a line by giving the slope and one point on the line. We give the first coordinate of some points on the line. Without deriving the equation of the line, find the second coordinate of each point.Slope is -3, (2, 2) on line; (3, ); (4, ); (1, ).
In Exercises, use limits to compute f′(x).f (x) = -x + 11
Estimate the cost of manufacturing 51 bicycles per day in Exercise 37.Exercise 37Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. Interpret C(50) = 5000 and C′(50) = 45.
If f(x) is a linear function, f(1) = 0, and f(2) = 1, what is f(3)?
Find the slope of the graph of y = f(x) at the designated point.f(x) = 3x2 - 2x + 1, (1, 2)
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the graph of f(x) = 1 Vx x
In Exercises, use limits to compute f′(x). f(x) = x + 1 X
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the graph of f(x) || +3⁹
In Exercises, you are given a function and a point on the graph of the function. Zoom in on the graph at the given point until it starts to look like a straight line. Estimate the slope of the graph at the point indicated. f(x) = x-1 x + 1 (1,0).
Find the slope of the graph of y = f(x) at the designated point. 10 f(x) = x¹0 + 1 + V1-x, (0, 2)
In Exercises, use limits to compute f′(x). fx) = 1 .2 X
In Exercises, you are given a function and a point on the graph of the function. Zoom in on the graph at the given point until it starts to look like a straight line. Estimate the slope of the graph at the point indicated.f (x) = 2x2 - 3x + 2, (0, 2).
For each pair of lines in the following figures, determine the one with the greater slope. 12- y X
If f(t) = 3t3 - 2t2, find f′(2).
The revenue from producing (and selling) x units of a product is given by R(x) = 3x - .01x2 dollars.(a) Find the marginal revenue at a production level of 20.(b) Find the production levels where the revenue is $200.
Is the line through the points (3, 4) and (-1, 2) parallel to the line 2x + 3y = 0? Justify your answer.
In Exercises, you are given a function and a point on the graph of the function. Zoom in on the graph at the given point until it starts to look like a straight line. Estimate the slope of the graph at the point indicated. f(x)=√x + 3, (1, 2).
Differentiate.If V(r) = 15πr2, find V′(1/3).
Let P(x) be the profit from producing (and selling) x units of goods. Match each question with the proper solution.A. What is the profit from producing 1000 units of goods?B. At what level of production will the marginal profit be 1000 dollars?C. What is the marginal profit from producing 1000
In Exercises, use limits to compute f′(x). f(x) = X x + 1
Let R(x) denote the revenue (in thousands of dollars) generated from the production of x units of computer chips per day, where each unit consists of 100 chips.(a) Represent the following statement by equations involving R or R′: When 1200 chips are produced per day, the revenue is $22,000 and
For each pair of lines in the following figures, determine the one with the greater slope. 41 y 12 X
The tangent to the graph of y = 1x at the point where a > 0, is perpendicular to the line y = 4x + 1. Find P. (a, a),
Find the slope of the tangent line to the curve y = x3 + 3x - 8 at (2, 6).
In Exercises, use limits to compute f′(x). f(x) = −1 + 2 x-2
The point–slope form of the equation of the tangent line to the graph of y = x4 at the point (1, 1) is y - 1 = 4(x - 1). Explain how this equation follows from formula (6).
Differentiate.If g(u) = 3u - 1, find g(5) and g′(5).
In Exercises, you are given a function and a point on the graph of the function. Zoom in on the graph at the given point until it starts to look like a straight line. Estimate the slope of the graph at the point indicated. f(x) = √x + 6, (2, 2).
Refer to Exercise 41. Is it profitable to produce 1300 chips per day if the cost of producing 1200 chips per day is $14,000?Exercise 41Let R(x) denote the revenue (in thousands of dollars) generated from the production of x units of computer chips per day, where each unit consists of 100 chips.
In Exercises, use limits to compute f′(x). f(x) = 1024 = 1 2 x² + 1 +1
Write the equation of the tangent line to the curve y = x3 + 3x - 8 at (2, 6).
(a) In Example 5, find the total sales for January 10, and determine the rate at which sales are falling on that day.(b) Compare the rate of change of sales on January 2 (Example 5) to the rate on January 10. What can you infer about the rate of change of sales?Example 5At the end of the holiday
In Exercises, use limits to compute f′(x). T + x X = (x)ƒ
Differentiate.If h(x) = - 1/2, find h(-2) and h′(-2).
Find the equation and sketch the graph of the following lines.With slope -2 and y-intercept (0, -1)
Let S(x) represent the total sales (in thousands of dollars) for the month x in the year 2005 at a certain department store. Represent each following statement by an equation involving S or S′. (a) The sales at the end of January reached $120,560 and were rising at the rate of $1500 per
Find the slope of the tangent line to the curve y = (x2 - 15)6 at x = 4. Then write the equation of this tangent line.
The line y = 2x + b is tangent to the graph of y = √x at the point P = (a, √a). Find P and determine b.
Differentiate.If f(x) = x5/2, what is f″(4)?
In Exercises, two lines intersect the graph of a function y = f(x) as shown in the figure. Find a and f (a). f(x) 0 y y = -x + 4 a y = 2 X
Find the equation and sketch the graph of the following lines.With slope 1/3 and y-intercept (0, 1)
Refer to Example 5.(a) Compute S(10) and S(10).(b) Use the data in part (a) to estimate the total sales on January 11. Compare your estimate to the actual value given by S(11)Example 5At the end of the holiday season in January, the sales at a department store are expected to fall (Fig. 3). It is
In Exercises, use limits to compute f′(x). て+ 7 + X^=(x)f
The financial analysts at the store in Example 5 corrected their projections and are now expecting the total sales for the x day of January to be(a) Let S(x) be as in Example 5. Compute T(1), T′(1), S(1), and S′(1).(b) Compare and interpret the data in part (a) as they pertain to the sales on
The line y = ax + b is tangent to the graph of y = x3 at the point P = (-3, -27). Find a and b.
Differentiate.If g(t) = 1/4(2t - 7)4, what is g″(3)?
In Exercises, use limits to compute f′(x). I + zx ^ = (x)ƒ
In Exercises, two lines intersect the graph of a function y = f(x) as shown in the figure. Find a and f (a). y 0 f(x) a y = x y = x + 1 X
Differentiate the function f(x) = (3x2 + x - 2)2 in two ways.(a) Use the general power rule.(b) Multiply 3x2 + x - 2 by itself and then differentiate the resulting polynomial.
Using the sum rule and the constant-multiple rule, show that for any functions f(x) and g(x) dx [ƒ(x) = g(x)] = f(x) - d dx - d dx 86 g(x).
(a) Find the point on the curve y = √x where the tangent line is parallel to the line y = x/8.(b) On the same axes, plot the curve y = √x, the line y = x/8, and the tangent line to y = √x that is parallel to y = x/8.
Find the slope of the graph of y = (3x - 1)3 - 4(3x - 1)2 at x = 0.
In Exercises, use limits to compute f′(x). f(x) = 1 Vx
Figure 2 contains the curves y = f(x) and y = g(x) and the tangent line to y = f(x) at x = 1, with g(x) = 3 · f(x). Find g(1) and g′(1). y y = .6x + 1 y = f(x) y = g(x) 1 Figure 2 Graphs of f(x) and g(x) = 3f(x). X
There are two points on the graph of y = x3 where the tangent lines are parallel to y = x. Find these points.
In Exercises, use limits to compute f′(x). f(x)=x√x
Find the slope of the graph of y = (4 - x)5 at x = 5.
Let C(x) = 12x + 1100 denote the total cost (in dollars) of manufacturing x units of a certain commodity per day.(a) What is the total cost if the production is set at 10 units per day?(b) What is the marginal cost?(c) Use (b) to determine the additional cost of raising the daily production level
(a) Let A(x) denote the number (in hundreds) of computers sold when x thousand dollars is spent on advertising. Represent the following statement by equations involving A or A′: When $8000 is spent on advertising, the number of computers sold is 1200 and is rising at the rate of 50 computers for
Figure 3 contains the curves y = f (x), y = g(x), and y = h(x) and the tangent lines to y = f (x) and y = g(x) at x = 1, with h(x) = f(x) + g(x). Find h(1) and h′(1). Y y = h(x) y = f(x) y = 4x + 2.6 y .26x + 1.1 1 Figure 3 Graphs of f(x), g(x), and h(x) = f(x) + g(x). y = g(x) X
Is there any point on the graph of y = x3 where the tangent line is perpendicular to y = x? Justify your answer.
Refer to Exercise 47. Use the formula for C(x) to show directly that C(x + 1) - C(x) = 12. Interpret your result as it pertains to the marginal cost.Exercise 47Let C(x) = 12x + 1100 denote the total cost (in dollars) of manufacturing x units of a certain commodity per day.
A toy company introduces a new video game on the market. Let S(x) denote the number of videos sold on the day, x, since the item was introduced. Let n be a positive integer. Interpret S(n), S′(n), and S(n) + S′(n).
The graph of y = f(x) goes through the point (2, 3) and the equation of the tangent line at that point is y = -2x + 7. Find f (2) and f′(2).
Each limit in Exercises is a definition of f′(a). Determine the function f(x) and the value of a. lim h→0 (1 + h)² - 1 2 h
In Exercises, find the indicated derivative. d dx (x8)
Compute the third derivatives of the following functions:(a) f(t) = t10(b) f(z) = 1 z +5
Each limit in Exercises is a definition of f′(a). Determine the function f(x) and the value of a. lim h→0 1 10+ h h -.1
The price of 1 gallon of unleaded gasoline at the pump dropped to $2.19 on January 1, 2015, and continued to fall at the rate of 4 cents per month for the next 9 months. Express the price of 1 gallon of unleaded gasoline as a function of time for the period starting January 1, 2015. What was the
Each limit in Exercises is a definition of f′(a). Determine the function f(x) and the value of a. lim h→0 (2+ h)³ - 8 h
The third derivative of a function f(x) is the derivative of the second derivative f″(x) and is denoted by f″′(x). Compute f″′(x) for the following functions:(a) f (x) = x5 - x4 + 3x(b) f (x) = 4x5/2
In Exercises, find the indicated derivative. d dx -(x-3)
In Exercises, find the indicated derivative. d dx -(x-³/4)
For the given function, simultaneously graph the functions f(x), f′(x), and f″(x) with the specified window setting. X f(x) = 1 + x2 [-4, 4] by [-2, 2].
The discovery of one case of bovine spongiform encephalopathy, or mad cow disease, in May 2003 in Canada led to an immediate ban on all Canadian beef exports. At the beginning of September 2003, the ban was lifted, and exports of Canadian beef rose at a steady rate of $42.5 million per month.
In Exercises, find the indicated derivative. d dx -(x-1/³)
An online bookstore charges $5 plus 3% of the purchase price of books for shipping and handling. Find a function C(x) that expresses the shipping and handling charge for a book order that costs x dollars.
Each limit in Exercises is a definition of f′(a). Determine the function f(x) and the value of a. lim h→0 (64+ h)¹/3 - 4 h
In Exercises, find the indicated derivative. dy - if y = 1 dx
Each limit in Exercises is a definition of f′(a). Determine the function f(x) and the value of a. lim h→0 √9+h-3 h
Each limit in Exercises is a definition of f′(a). Determine the function f(x) and the value of a. lim h→0 (1 +h)-¹/2-1 h V
In industry, the relationship between wages and the quit ratio of employees is defined to be the percentage of employees that quit within 1 year of employment. The quit ratio of a large restaurant chain that paid its employees the minimum hourly wage ($7.25 per hour) was .2 or 20 employees per 100.
Consider the cost function of Example 6.(a) Graph C(x) in the window [0, 60] by [-300, 1260].(b) For what level of production will the cost be $535?(c) For what level of production will the marginal cost be $14?Example 6.Suppose that the cost of producing x items isC(x) = .005x3 - .5x2 + 28x + 300
When the owner of a gas station sets the price of 1 gallon of unleaded gasoline at $2.10, she can sell approximately 1500 gallons per day. When she sets the price at $2.25 per gallon, she can sell approximately 1250 gallons per day. Let G(x) denote the number of gallons of unleaded gasoline sold
Compute the following limits. lim 1 Xxx
In Exercises, find the indicated derivative. dy - if y = x-4 dx
In Exercises, find the indicated derivative. dy dx - if y = x¹/5
The tangent line to the curve y = 1/3x3 - 4x2 + 18x + 22 is parallel to the line 6x - 2y = 1 at two points on the curve. Find the two points.
The straight line in the figure is tangent to the graph of f(x). Find f(4) and f′(4). 5 (0, 3) Y 4 y = f(x) X
Refer to Exercise 53. Where should the owner set the price if she wants to sell 2200 gallons per day?Exercise 53When the owner of a gas station sets the price of 1 gallon of unleaded gasoline at $2.10, she can sell approximately 1500 gallons per day. When she sets the price at $2.25 per gallon, she

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