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mathematics
applied calculus

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

The tangent line to the curve y = x3 - 6x2 - 34x - 9 has slope 2 at two points on the curve. Find the two points.
The straight line in the figure is tangent to the parabola. Find the value of b. y = x² - 4x + 10 (0, b) y 6 X
A company manufactures and sells fishing rods. The company has a fixed cost of $1500 per day and a total cost of $2200 per day when the production is set at 100 rods per day. Assume that the total cost C(x) is linearly related to the daily production level x.(a) Express the total cost as a function
A salesperson’s weekly pay depends on the volume of sales. If she sells x units of goods, her pay is y = 5x + 60 dollars. Give an interpretation of the slope and the y-intercept of this straight line.
Compute the following limits. lim 1 ざ .2 x8-x
In Exercises, find the indicated derivative. 3 I- x if y xp ap
Consider the curve y = f (x) in Fig. 13. Find f (6) and f′(6). y Figure 13 6 y = f(x) Y +2 (tangent line) =
Compute the following limits. lim xx 5x+3 3x - 2
Consider the curve y = f (x) in Fig. 14. Find f (1) and f′(1). 1 y = f(x) y = 4 (tangent line) X
Compute the following limits. lim 1 8-xx-x
The demand equation for a manufacturer is y = -.02x + 7, where x is the number of units produced and y is the price. That is, to sell x units of goods, the price must be y = -.02x + $7. Interpret the slope and y intercept of this line.
In Fig. 15, the straight line y = 14x + b is tangent to the graph of f(x) = 1x. Find the values of a and b. Y y = = x + b Figure 15 a y = √x X
Temperatures of 32°F and 212°F correspond to temperatures of 0°C and 100°C. The linear equation y = mx + b converts Fahrenheit temperatures to Celsius temperatures. Find m and b. What is the Celsius equivalent of 98.6°F?
Compute the following limits. lim x- X→∞ 10x + 100 x² - 30 2 X
Find the equation of the tangent line to the curve y = x2 at the pointSketch the graph of y = x2 and sketch the tangent line at 2, 4
What is the slope of the graph of f (x) = x3 - 4x2 + 6 at x = 2? Write the equation of the line tangent to the graph of f(x) at x = 2.
A drug is administered to a patient through an IV (intravenous) injection at the rate of 6 milliliters (mL) per minute. Assuming that the patient’s body already contained 1.5 mL of this drug at the beginning of the infusion, find an expression for the amount of the drug in the body x minutes from
Compute the following limits. x + zx lim xx x² - 1
In Fig. 16, the straight line is tangent to the graph of f(x) = 1/x. Find the value of a. y y ==// X Figure 16 2 a X
What is the slope of the curve y = 1/(3x - 5) at x = 1? Write the equation of the line tangent to this curve at x = 1.
In Exercises, refer to Fig. 5 to find the given limit. 1 0 Figure 5 f(x) 1 8
Refer to Exercise 59. If the patient’s body eliminates the drug at the rate of 2 mL per hour, find an expression for the amount of the drug in the body x minutes from the start of the infusion.Exercise 59A drug is administered to a patient through an IV (intravenous) injection at the rate of 6
Consider the curve y = f(x) in Fig. 17. Find a and f (a). Estimate f(a). a Figure 17 y = f(x) y = 2.03x.53 y = 2.02x.52 y = 2.01x .51 - X
Consider the curve y = f(x) in Fig. 18. Estimate f′(1). 1 Y 0 Figure 18 (1.4, 1.3) (1.2, 1.1), (1,.8) (1.6, 1.4) y = f(x) 2 -X
In Exercises, refer to Fig. 5 to find the given limit. 1 0 Figure 5 f(x) 1 8
After inspecting a sunken ship at a depth of 212 feet, a diver starts her slow ascent to the surface of the ocean, rising at the rate of 2 feet per second. Find y(t), the depth of the diver, measured in feet from the ocean’s surface, as a function of time t (in seconds).
In Exercises, refer to Fig. 5 to find the given limit. 1 0 Figure 5 f(x) 1 8
In Fig. 19, find the equation of the tangent line to f(x) at the point A. 6 LO 5 4 3 2 1 Y A y = f(x) 2 4₁ 6 Figure 19 8 10 12 14 X
Find the equation of the tangent line to the curve y = x2 at the point (-2, 4). Sketch the graph of y = x2 and sketch the tangent line at (-2, 4).
The diver in the previous exercise is supposed to stop for 5 minutes and decompress at 150 feet depth. Assuming that the diver will continue her ascent, after decompressing, at the same rate of 2 feet per second, find y(t) in this case and determine how long it will take the diver to reach the
In Exercises, refer to Fig. 5 to find the given limit. 1 0 Figure 5 f(x) 1 8
In Fig. 1, the straight line has slope -1 and is tangent to the graph of f(x). Find f(2) and f′(2). m = -1 Figure 1 2 5 y = f(x) X
Determine the equation of the tangent line to the curve y = 3x3 - 5x2 + x + 3 at x = 1.
In Fig. 20, find the equation of the tangent line to f(x) at the point P. CO 3 2 Y Figure 20 P y = f(x) y = f(x) HH X
A T-shirt shop owner has a fixed cost of $230 and a marginal cost of $7 per T-shirt to manufacture x T-shirts per day. Let C(x) denote the cost to manufacture x T-shirts per day.(a) Find C(x).(b) If the shop owner decides to sell the T-shirts at $12 each, find R(x), the total revenue from selling x
In Exercises, refer to Fig. 5 to find the given limit. 1 0 Figure 5 f(x) 1 8
In Exercises, compute the difference quotientSimplify your answer as much as possible.f(x) = 2x2 f(x +h)-f(x) h
In Fig. 2, the straight line is tangent to the graph of f (x) = x3. Find the value of a. a Figure 2 (0, 2) y X
If, for some constant m,for all x1 ≠ x2, show that f(x) = mx + b, where b is some constant. f(x₂)-f(x₁) = m m X2X1
Determine the equation of the tangent line to the curve y = (2x2 - 3x)3 at x = 2.
In Exercises, refer to Fig. 5 to find the given limit. 1 0 Figure 5 f(x) 1 8
In Exercises, compute the difference quotientSimplify your answer as much as possible.f (x) = x2 - 7 f(x +h)-f(x) h
In order for a business to break even, revenue has to equal cost. Determine the minimum number of T-shirts that should be sold in the previous exercise to break even.
Examine the graph of the function and evaluate the function-atlarge values of x to guess the value of the limit. zx lim x→∞ 2x X
Examine the graph of the function and evaluate the function-atlarge values of x to guess the value of the limit. lim √25 + x - √x
In Exercises, compute the difference quotientSimplify your answer as much as possible.f (x) = -x2 + 2x f(x +h)-f(x) h
In Exercises, compute the difference quotientSimplify your answer as much as possible.f (x) = -2x2 + x + 3 f(x +h)-f(x) h
(a) Draw the graph of any function f(x) that passes through the point (3, 2).(b) Choose a point to the right of x = 3 on the x-axis and label it 3 + h.(c) Draw the straight line through the points (3, f (3)) and (3 + h, f (3 + h)).(d) What is the slope of this straight line (in terms of h)?
Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path.How far has the person traveled after 6 seconds? Y 12 11 10 9 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5 6 7 Figure 3 Walker's progress. t
A helicopter is rising at a rate of 32 feet per second. At a height of 128 feet the pilot drops a pair of binoculars. After t seconds, the binoculars have height s(t) = -16t2 + 32t + 128 feet from the ground. How fast will they be falling when they hit the ground?
Let y denote the percentage of the world population that is urban x years after 2014. According to data from the United Nations, 54 percent of the world’s population was urban in 2014, and projections show that this percentage will increase to 66 percent by 2050. Assume that y is a linear
Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path.What is the person’s average velocity from time t = 1 to t = 4? Y 12 11 10 9 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5 6 7 Figure 3 Walker's progress. t
Examine the graph of the function and evaluate the function-atlarge values of x to guess the value of the limit. lim x² - 2x + 3 2x² + 1
Each day the total output of a coal mine after t hours of operation is approximately 40t + t2 - 1/15t3 tons, 0 ≤ t ≤ 12. What is the rate of output (in tons of coal per hour) at t = 5 hours?
Let y denote the average amount claimed for itemized deductions on a tax return reporting x dollars of income. According to Internal Revenue Service data, y is a linear function of x. Moreover, in a recent year income tax returns reporting $20,000 of income averaged $729 in itemized deductions,
In Exercises, compute the difference quotient Simplify your answer as much as possible.f(x) = x3 (a + b)3 = a3 + 3a2b + 3ab2 + b3. f(x +h)-f(x) h
Examine the graph of the function and evaluate the function-atlarge values of x to guess the value of the limit. lim -X -8x² + 1 2 2 x² + 1
Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path.Without calculating velocities, determine whether the person is traveling faster at t = 5 or at t = 6. Y 12 11 10 9 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5 6 7 Figure 3 Walker's
In Exercises, compute the difference quotientSimplify your answer as much as possible.f (x) = 2x3 + x2 f(x +h)-f(x) h
Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path.What is the person’s velocity at time t = 3? Y 12 11 10 9 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5 6 7 Figure 3 Walker's progress. t
In Exercises, apply the three-step method to compute the derivative of the given function. f (x) = -x2
In Exercises, apply the three-step method to compute the derivative of the given function.f(x) = 3x2 - 2
(a) Draw two graphs of your choice that represent a function y = f (x) and its vertical shift y = f (x) + 3.(b) Pick a value of x and consider the points (x, f (x)) and (x, f (x) + 3). Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines.(c)
In Exercises, apply the three-step method to compute the derivative of the given function.f (x) = 7x2 + x - 1
A manufacturer estimates that the hourly cost of producing x units of a product on an assembly line is C(x) = .1x3 - 6x2 + 136x + 200 dollars.(a) Compute C(21) - C(20), the extra cost of raising the production from 20 to 21 units.(b) Find the marginal cost when the production level is 20 units.
In Exercises, apply the three-step method to compute the derivative of the given function.f (x) = x + 3
The number of people riding the subway daily from Silver Spring, Maryland, to Washington’s Metro Center is a function f(x) of the fare, x cents. If f(235) = 4600 and f′(235) = -100, approximate the daily number of riders for each of the following costs:(a) 237 cents (b) 234 cents(c) 240
Determine whether the following limits exist. If so, compute the limit. - 4 lim x 2 x 2 -
In Exercises, apply the three-step method to compute the derivative of the given function.f (x) = x3
Determine whether the following limits exist. If so, compute the limit. 1 lim x-3x² - 4x +3 2
Let h(t) be a boy’s height (in inches) after t years. If h′(12) = 1.5, how much will his height increase (approximately) between ages 12 and 12(1/2)?
In Exercises, use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. f'(0), where f(x) = 2x
In Exercises, apply the three-step method to compute the derivative of the given function.f (x) = 2x3 - x
Determine whether the following limits exist. If so, compute the limit. x - 4 lim x-4x²8x + 16 2
If you deposit $100 in a savings account at the end of each month for 2 years, the balance will be a function f(r) of the interest rate, r%. At 7% interest (compounded monthly), f(7) = 2568.10 and f′(7) = 25.06. Approximately how much additional money would you earn if the bank paid 7(12)%
Determine whether the following limits exist. If so, compute the limit. x-5 lim x 5x²7x + 2 2
In Exercises, use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. f'(1), where f(x) = 1 1+x²
In Exercises, use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. f'(1), where f(x) = V1 + x²
In Exercises, use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. f'(2), where f(x) 1 + x
What geometric interpretation may be given toin connection with the graph of f (x) = x2? (3 + h)² - 3² h 32
In Exercises, use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. f'(3), where f(x) = V25 - x²
In Exercises, use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. f'(0), where f(x) = 10¹+x
Use limits to compute the following derivatives.f′(5), where f(x) = 1/(2x).
As h approaches 0, what value is approached by 1 2+h h 1 2 -?
Use limits to compute the following derivatives.f′(3), where f(x) = x2 - 2x + 1.
A company finds that the revenue R generated by spending x dollars on advertising is given by dR R = 1000 + 80x - .02x², for 0 ≤ x ≤ 2000. Find dx x = 1500
Each of lines (A), (B), (C), and (D) in the figure is the graph of one of the equations (a), (b), (c), and (d). Match each equation with its graph.(a) x + y = 1 (b) x - y = 1(c) x + y = -1 (d) x - y = -1 Y X
National health expenditures (in billions of dollars) from 1980 to 1998 are given by the function f(t) in Fig. 8.(a) How much money was spent in 1987?(b) Approximately how fast were expenditures rising in 1987?(c) When did expenditures reach $1 trillion?(d) When were expenditures rising at the rate
In the next section, we shall see that the tangent line to the graph of y = x3 at the point (x, y) has slope 3x2. See Fig. 15. Using this result, find the slope of the curve at the points in Exercises.Figure 15 Slope of tangent line to y = x3. Y y = x³ (x, y) Slope is 3x2 X
The graph in Fig. 8 shows the total sales in thousands of dollars in a department store during a typical 24-hour period.(a) Estimate the rate of sales during the period between 8 a.m. and 10 a.m.(b) Which 2-hour interval in the day sees the highest rate of sales and what is this rate? Sales in
Differentiate.f(x) = [x5 - (x - 1)5]10
In the next section, we shall see that the tangent line to the graph of y = x3 at the point (x, y) has slope 3x2. See Fig. 15. Using this result, find the slope of the curve at the points in Exercises.Figure 15 Slope of tangent line to y = x3. Y y = x³ (x, y) Slope is 3x2 X
Estimate how much the functionwill change if x decreases from 1 to .9. f(x) = 1 1 + x²
Compute the following. d (dv dt dt where v = 21² + 1 t + 1
If f(x) = 1/x5, compute f (-2) and f′(-2).
Use limits to compute the following derivatives.f′(0), where f(x) = x3 + 3x + 1
In Exercises, we specify a line by giving the slope and one point on the line. Start at the given point and use plotting a line using a slope and a point to sketch the graph of the line.m = 0, (0, 2) on line
Compute the following. d dv dt dt t=2 where v(t) = 31³ + 4 t
Do Exercise 29 if it costs 10 cents per copy for the first 50 copies and 5 cents per copy for each copy exceeding 50, and there is no setup fee.Exercise 29The owner of a photocopy store charges 7 cents per copy for the first 100 copies and 4 cents per copy for each copy exceeding 100. In addition,
If f(x) = 1/x2, compute f (1) and f′(1).

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