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mathematics
calculus early transcendentals

Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions

Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, it has been determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given byC(x)
Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67 + 0.70b + 0.0067b2.a. Graph the height function.b. Plot and
The position (in meters) of a marble rolling up a long incline is given by s = 100t / t + 1, where t is measured in seconds and s = 0 is the starting point.a. Graph the position function.b. Find the velocity function for the marble.c. Graph the velocity function and give a description of the motion
Economists use production functions to describe how the output of a system varies with respect to another variable such as labor or capital. For example, the production function P(L) = 200L + 10L2 - L3 gives the output of a system as a function of the number of laborers L. The average product A(L)
Suppose p(t) represents the population of the United States (in millions) t years after the year 1900. The graph of the growth rate p' is shown in the figure.a. Approximately when (in what year) was the U.S. population growing most slowly between 1900 to 1990? Estimate the growth rate in that
Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal
Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal
Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal
Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal
Assume the length L (in cm) of a particular species of fish after t years is modeled by the following graph.a. What does dL/dt represent and what happens to this derivative as t increases?b. What does the derivative tell you about how this species of fish grows?c. Sketch a graph of L' and L". LA
The graph of s = f(t) represents the position of an object moving along a line at time t ≥ 0.a. Assume the velocity of the object is 0 when t = 0. For what other values of t is the velocity of the object zero?b. When is the object moving in the positive direction and when is it moving in the
The graph shows the position s = f(t) of a car t hours after 5:00 p.m. relative to its starting point s = 0, where s is measured in miles.a. Describe the velocity of the car. Specifically, when is it speeding up and when is it slowing down?b. At approximately what time is the car traveling the
A stone is thrown from the edge of a bridge that is 48 ft above the ground with an initial velocity of 32 ft/s. The height of this stone above the ground t seconds after it is thrown is f (t) = -16t2 + 32t + 48. If a second stone is thrown from the ground, then its height above the ground after t
Two stones are thrown vertically upward with matching initial velocities of 48 ft/s at time t = 0. One stone is thrown from the edge of a bridge that is 32 ft above the ground and the other stone is thrown from ground level. The height of the stone thrown from the bridge after t seconds is f(t) =
A stone is thrown vertically into the air at an initial velocity of 96 ft/s. On Mars, the height s (in feet) of the stone above the ground after t seconds is s = 96t - 6t2 and on Earth it is s = 96t - 16t2. How much higher will the stone travel on Mars than on Earth?
On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Then its height s (in meters) above the ground after t seconds is s = 40 - 0.8t2. Determine the velocity and acceleration of the
Determine whether the following statements are true and give an explanation or counterexample.a. If the acceleration of an object remains constant, then its velocity is constant.b. If the acceleration of an object moving along a line is always 0, then its velocity is constant.c. It is impossible
Show that the demand function D(p) = a/pb, where a and b are positive real numbers, has a constant elasticity for all positive prices.
Compute the elasticity for the exponential demand function D(p) = ae-bp, where a and b are positive real numbers. For what prices is the demand elastic? Inelastic?
The economic advisor of a large tire store proposes the demand function D(p) = 1800 / p - 40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.a. Recalling that the demand must be positive, what is the domain of this function?b. According to the
Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.a. According to the model, how many DVDs can be sold in a day at price of $10?b. According to the model,
Consider the following cost functions.a. Find the average cost and marginal cost functions.b. Determine the average and marginal cost when x = a.c. Interpret the values obtained in part (b).C(x) = -0.04x2 + 100x + 800, 0 ≤ x ≤ 1000, a = 500
Consider the following cost functions.a. Find the average cost and marginal cost functions.b. Determine the average and marginal cost when x = a.c. Interpret the values obtained in part (b).C(x) = -0.01x2 + 40x + 100, 0 ≤ x ≤ 1500, a = 1000
Consider the following cost functions.a. Find the average cost and marginal cost functions.b. Determine the average and marginal cost when x = a.c. Interpret the values obtained in part (b).C(x) = 500 + 0.02x, 0 ≤ x ≤ 2000, a = 1000
Consider the following cost functions.a. Find the average cost and marginal cost functions.b. Determine the average and marginal cost when x = a.c. Interpret the values obtained in part (b).C(x) = 1000 + 0.1x, 0 ≤ x ≤ 5000, a = 2000
The U.S. consumer price index (CPI) measures the cost of living based on a value of 100 in the years 1982–1984. The CPI for the years 1995–2012 (see figure) is modeled by the function c(t) = 151e0.026t, where t represents years after 1995.a. Was the average growth rate of the CPI greater
The population of the state of Georgia (in thousands) from 1995 1t = 02 to 2005 (t = 10) is modeled by the polynomial p(t) = -0.27t2 + 101t + 7055.a. Determine the average growth rate from 1995 to 2005.b. What was the growth rate for Georgia in 1997 (t = 2) and 2005 (t = 10)?c. Use a graphing
Suppose a stone is thrown vertically upward from the edge of a cliff on Mars (where the acceleration due to gravity is only about 12 ft/s2)with an initial velocity of 64 ft/s from a height of 192 ft above the ground. The height s of the stone above the ground after t seconds is given by s = -6t2 +
Suppose a stone is thrown vertically upward from the edge of a cliff with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height s (in ft) of the stone above the ground t seconds after it is thrown is s = -16t2 + 64t + 32.a. Determine the velocity v of the stone after t
Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.a. Graph the position function.b. Find and graph the velocity function. When is the object
Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.a. Graph the position function.b. Find and graph the velocity function. When is the object
Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.a. Graph the position function.b. Find and graph the velocity function. When is the object
Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.a. Graph the position function.b. Find and graph the velocity function. When is the object
Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.a. Graph the position function.b. Find and graph the velocity function. When is the object
Suppose the position of an object moving horizontally after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.a. Graph the position function.b. Find and graph the velocity function. When is the object
The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 a.m. The plane returns to Seattle 8.5 hours later at 2:30 p.m.a. Calculate the average
A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 a.m. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 a.m. (see figure). Assume s is positive when the car is north of the patrol station.a.
Explain why a decreasing demand function has a negative elasticity function.
Suppose the average cost of producing 200 gas stoves is $70 per stove and the marginal cost at x = 200 is $65 per stove. Interpret these costs.
An object moving along a line has a constant negative acceleration. Describe the velocity of the object.
Define the acceleration of an object moving in a straight line.
What is the difference between the velocity and peed of an object moving in a straight line?
Complete the following statement: If dy/dx is small, then small changes in x result in relatively ________ changes in the value of y.
Complete the following statement. If dy/dx is large, then small changes in x result in relatively ________ changes in the value of y.
Explain the difference between the average rate of change and the instantaneous rate of change of a function f.
Suppose that f is differentiable for all x and consider the functionFor the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?f(x) = x3/3 + 1 on [-2, 2] f(x + 0.01) – f(x) D(x) 0.01
Suppose that f is differentiable for all x and consider the functionFor the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?f(x) = sin x on [-π, π] f(x + 0.01) – f(x) D(x) 0.01
The following limits equal the derivative of a function f at a point a.a. Find one possible f and a.b. Evaluate the limit. tan ( 5п (* + h) + V3 lim h→0
The following limits equal the derivative of a function f at a point a.a. Find one possible f and a.b. Evaluate the limit. 1 cot x lim x—п/4 х — TT
The following limits equal the derivative of a function f at a point a.a. Find one possible f and a.b. Evaluate the limit. V3 cos (7 + h) lim |h→0 – 2
The following limits equal the derivative of a function f at a point a.a. Find one possible f and a.b. Evaluate the limit. sin (7 + h) lim 2
Prove thatd2n/dx2n (sin x) = (-1)n sin x and d2n/dx2n (cos x) = (-1)n cos x.
Calculate the following derivatives using the Product Rule.a. d/dx (sin2 x)b. d/dx (sin3 x)c. d/dx (sin4 x)d. Based on your answers to parts (a)–(c), make a conjecture about d/dx (sinn x), where n is a positive integer. Then prove the result by induction.
Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.a. Explain why s(x) = sin (π/180 x)b. EvaluateVerify your answer by estimating the limit on your
Let For what values of a is f continuous? 3 sin x if x + 0 f(x) . if x = 0.
Use the definition of the derivative and the trigonometric identitycos (x + h) = cos x cos h - sin x sin hto prove that d/dx (cos x) = -sin x.
Use the half-angle formulato prove that 1 - cos 2x sinx cos x lim 0.
Use the trigonometric identity cos2 x + sin2 x = 1 to prove that Begin by multiplying the numerator and denominator by cos x + 1. - 1 = 0. cos x lim
Use the identity sin 2x = 2 sin x cos x to find d/dx (sin 2x). Then use the identity cos 2x = cos2 x - sin2 x to express the derivative of sin 2x in terms of cos 2x.
A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y"(t) + y(t) = 0.a. Show that y = A sin t satisfies the equation for any constant A.b. Show that y = B cos t satisfies the equation for any constant B.c. Show that y = A sin
The graph of f (t) = e-kt sin t with k > 0 is called a damped sine wave; it is used in a variety of applications, such as modeling the vibrations of a shock absorber.a. Use a graphing utility to graph f for k = 1, 1/2 , and 1/10 to understand why these curves are called damped sine waves. What
An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30 (sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.a. Graph the position function, for 0 ≤ t ≤ 10.b. Find the velocity of the oscillator, v(t) = y'(t).c.
Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. Functions a-d (b) (d) Derivatives A-D х UUJUIUI (A) (B) -27 (D)
For what values of x does f(x) = x - 2 cos x have a horizontal tangent line?
a. For what values of x does g(x) = x - sin x have a horizontal tangent line?b. For what values of x does g(x) = x - sin x have a slope of 1?
a. Find an equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line.y = cos x / 1 - cos x; x = π/3
a. Find an equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line.y = csc x; x = π/4
a. Find an equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line.y = 1 + 2 sin x; x = π/6
a. Find an equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line.y = 4 sin x cos x; x = π/3
Find dy/dx for the following functions.y = 1 - cos x / 1 + cos x
Find dy/dx for the following functions.y = x cos x / 1 + x3
Find dy/dx for the following functions.y = sin x / sin x - cos x
Find dy/dx for the following functions.y = 1 / 2 + sin x
Find dy/dx for the following functions.y = x cos x sin x
Find dy/dx for the following functions.y = sin x / 1 + cos x
Evaluate the following limits or state that they do not exist. lim 3 csc 2x cot 2x x→T/4
Evaluate the following limits or state that they do not exist. cos x lim 00 х
Evaluate the following limits or state that they do not exist. 3 sec’x So lim .2 x→0 x- + 4 - 4
Evaluate the following limits or state that they do not exist. cos X lim TT/2 X (T/2)
Evaluate the following limits or state that they do not exist.where a and b are constants with b ≠ 0 sin ax S1 lim x→0 sin bx'
Evaluate the following limits or state that they do not exist.where a and b are constants with b ≠ 0 sin ax lim x→0 bx
Determine whether the following statements are true and give an explanation or counterexample.a d/dx (sin2 x) = cos2 x.b. d2/dx2 (sin x) = sin x.c. d4/dx4 (cos x) = cos x.d. The function sec x is not differentiable at x = π/2.
Find y" for the following functions.y = cos θ sin θ
Find y" for the following functions.y = sec x csc x
Find y" for the following functions.y = tan x
Find y" for the following functions.y = cot x
Find y" for the following functions.y = 1/2 ex cos x
Find y" for the following functions.y = ex sin x
Find y" for the following functions.y = cos x
Find y" for the following functions.y = x sin x
Find the derivative of the following functions.y = csc2 θ - 1
Find the derivative of the following functions.y = 1 / sec z csc z
Find the derivative of the following functions.y = tan t / 1 + sec t
Find the derivative of the following functions.y = cot x / 1 + csc x
Find the derivative of the following functions.y = tan w / 1 + tan w
Find the derivative of the following functions.y = e5x csc x
Find the derivative of the following functions.y = sec x tan x
Find the derivative of the following functions.y = sec x + csc x
Find the derivative of the following functions.y = tan x + cot x
Verify the following derivative formulas using the Quotient Rule.d/dx (csc x) = -csc x cot x

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