A manufacturers cost of producing a product is given in Figure 4.120. The manufacturer

For Exercises 12, indicate all critical points on the given graphs. Which correspond to local minima, local maxima, global maxima, global minima, or none of these? (Note that the graphs are on closed intervals.)

For Exercises 12, indicate all critical points on the given graphs. Which correspond to local minima, local maxima, global maxima, global minima, or none of these? (Note that the graphs are on closed intervals.)

In Exercises 36, do the following: (a) Find f and f. (b) Find the critical points of f. (c) Find any inflection points. (d) Evaluate f at the critical points and the endpoints. Identify the global maxima and minima of f. (e) Sketch f. Indicate clearly where f is increasing or decreasing, and its concavity

In Exercises 36, do the following: (a) Find f and f. (b) Find the critical points of f. (c) Find any inflection points. (d) Evaluate f at the critical points and the endpoints. Identify the global maxima and minima of f. (e) Sketch f. Indicate clearly where f is increasing or decreasing, and its concavity

In Exercises 36, do the following: (a) Find f and f. (b) Find the critical points of f. (c) Find any inflection points. (d) Evaluate f at the critical points and the endpoints. Identify the global maxima and minima of f. (e) Sketch f. Indicate clearly where f is increasing or decreasing, and its concavity

In Exercises 36, do the following: (a) Find f and f. (b) Find the critical points of f. (c) Find any inflection points. (d) Evaluate f at the critical points and the endpoints. Identify the global maxima and minima of f. (e) Sketch f. Indicate clearly where f is increasing or decreasing, and its concavity

In Exercises 79, find the limits as x tends to + and , and then proceed as in Exercises 36. (That is, find f , etc.)

In Exercises 79, find the limits as x tends to + and , and then proceed as in Exercises 36. (That is, find f , etc.)

In Exercises 79, find the limits as x tends to + and , and then proceed as in Exercises 36. (That is, find f , etc.)

In Exercises 1013, find the global maximum and minimum for the function on the closed interval

In Exercises 1013, find the global maximum and minimum for the function on the closed interval

In Exercises 1013, find the global maximum and minimum for the function on the closed interval

In Exercises 1013, find the global maximum and minimum for the function on the closed interval

In Exercises 1416, find the exact global maximum and minimum values of the function.

In Exercises 1416, find the exact global maximum and minimum values of the function.

In Exercises 1416, find the exact global maximum and minimum values of the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

In Exercises 1723, use derivatives to identify local maxima and minima and points of inflection. Graph the function.

Find the point where the following curve is steepest: y = 50 1+6e2t for t 0.

The graphs of the function f(x) = x/(x2 + a2) for a = 1, 2 , and 3, are shown in Figure 4.113. Without a calculator, identify the graphs by locating the critical points of f(x).

The graphs of the function f(x)=1eax for a = 1, 2, and 3, are shown in Figure 4.114. Without a calculator, identify the graphs.

(a) Find all critical points and all inflection points of the function f(x) = x4 2ax2 +b. Assume a and b are positive constants. (b) Find values of the parameters a and b if f has a critical point at the point (2, 5). (c) If there is a critical point at (2, 5), where are the inflection points?

(a) For a a positive constant, find all critical points of f(x) = x a x. (b) What value of a gives a critical point at x = 5? Does f(x) have a local maximum or a local minimum at this critical point?

If a and b are nonzero constants, find the domain and all critical points of f(x) = ax2 x b .

The average of two nonnegative numbers is 180. What is the largest possible product of these two numbers?

The product of three positive numbers is 192, and one of the numbers is twice one of the other numbers. What is the minimum value of their sum?

The difference between two numbers is 24. If both numbers are 100 or greater, what is the minimum value of their product?

(a) Fixed costs are $3 million; variable costs are $0.4 million per item. Write a formula for total cost as a function of quantity, q. (b) The item in part (a) is sold for $0.5 million each. Write a formula for revenue as a function of q. (c) Write a formula for the profit function for this item

A number x is increasing. When x = 10, the square of x is increasing at 5 units per second. How fast is x increasing at that time?

The mass of a cube in grams is M = x3 + 0.1x4, where x is the length of one side in centimeters. If the length is increasing at a rate of 0.02 cm/hr, at what rate is the mass of the cube increasing when its length is 5 cm?

If is the angle between a line through the origin and the positive x-axis, the area, in cm 2, of part of a cardioid is A = 3 + 4 sin + 1 2 sin(2). If the angle is increasing at a rate of 0.3 radians per minute, at what rate is the area changing when = /2?

In Exercises 3738, describe the motion of a particle moving according to the parametric equations. Find an equation of the curve along which the particle moves

In Exercises 3738, describe the motion of a particle moving according to the parametric equations. Find an equation of the curve along which the particle moves

Figure 4.115 is the graph of f , the derivative of a function f. At which of the points 0, x1, x2, x3, x4, x5, is the function f: (a) At a local maximum value? (b) At a local minimum value? (c) Climbing fastest? (d) Falling most steeply?

Figure 4.116 is a graph of f . For what values of x does f have a local maximum? A local minimum?

On the graph of f in Figure 4.117, indicate the x-values that are critical points of the function f itself. Are they local maxima, local minima, or neither?

Graph f given that: f (x)=0 at x = 2, f (x) < 0 for x < 2, f (x) > 0 for x > 2, f(x)=0 at x = 4, f(x) > 0 for x < 4, f(x) < 0 for x > 4.

A cubic polynomial with a local maximum at x = 1, a local minimum at x = 3, a y-intercept of 5, and an x3 term whose coefficient is 1

A quartic polynomial whose graph is symmetric about the y-axis and has local maxima at (1, 4) and (1, 4) and a y-intercept of 3.

A function of the form y = axb ln x, where a and b are nonzero constants, which has a local maximum at the point (e2, 6e1).

A function of the form y = A sin(Bx) + C with a maximum at (5, 2), a minimum at (15, 1.5), and no critical points between these two points.

A function of the form y = axebx2 with a global maximum at (1, 2) and a global minimum at (1, 2).

A closed rectangular box, with a square base x by x cm and height h cm.

A open-topped rectangular box, with a square base x by x cm and height h cm.

A closed cylinder with radius r cm and height h cm.

A cylinder open at one end with radius r cm and height h cm

In Problems 5254, find the best possible bounds for the functions

In Problems 5254, find the best possible bounds for the functions

In Problems 5254, find the best possible bounds for the functions

Find the value(s) of m, if any, that give the global maximum and minimum of P as a function of m where 6jm2+P = 4jk5km, for positive constants j and k.

Find values of a and b so that the function y = axebx has a local maximum at the point (2, 10)

(a) Find all critical points of f(t) = at2ebt, assuming a and b are nonzero constants. (b) Find values of the parameters a and b so that f has a critical point at the point (5, 12). (c) Identify each critical point as a local maximum or local minimum of the function f in part (b)

What effect does increasing the value of a have on the graph of f(x) = x2 + 2ax? Consider roots, maxima and minima, and both positive and negative values of a.

Sketch several members of the family y = x3 ax2 on the same axes. Show that the critical points lie on the curve y = 1 2 x3.

Sketch several members of the family y = x3 ax2 on the same axes. Show that the critical points lie on the curve y = 1 2 x3.

An object at a distance p from a thin glass lens produces an image at a distance q from the lens, where 1 p + 1 q = 1 f . The constant f is called the focal length of the lens. Suppose an object is moving at 3 mm per second toward a lens of focal length 15 mm. How fast and in what direction is the image moving at the moment at which the object is 35 mm from the lens

Any body radiates energy at various wavelengths. Figure 4.118 shows the intensity of the radiation of a black body at a temperature T = 3000 kelvins as a function of the wavelength. The intensity of the radiation is highest in the infrared range, that is, at wavelengths longer than that of visible light (0.40.7m). Max Plancks radiation law, announced to the Berlin Physical Society on October 19, 1900, states that r() = a 5(eb/ 1). Find constants a and b so that the formula fits the graph. (Later in 1900 Planck showed from theory that a = 2c2h and b = hc T k where c = speed of light, h = Plancks constant, and k = Boltzmanns constant.)

An electric current, I, in amps, is given by I = cos(wt) + 3 sin(wt), where w = 0 is a constant. What are the maximum and minimum values of I?

The efficiency of a screw, E, is given by E = ( 2) + , > 0, where is the angle of pitch of the thread and is the coefficient of friction of the material, a (positive) constant. What value of maximizes E?

A rectangle has one side on the x-axis and two corners on the top half of the circle of radius 1 centered at the origin. Find the maximum area of such a rectangle. What are the coordinates of its vertices?

The hypotenuse of a right triangle has one end at the origin and one end on the curve y = x2e3x, with x 0. One of the other two sides is on the x-axis, the other side is parallel to the y-axis. Find the maximum area of such a triangle. At what x-value does it occur?

Which point on the parabola y = x2 is nearest to (1, 0)? Find the coordinates to two decimals. [Hint: Minimize the square of the distancethis avoids square roots.]

Find the coordinates of the point on the parabola y = x2 which is closest to the point (3,0).

The cross-section of a tunnel is a rectangle of height h surmounted by a semicircular roof section of radius r (See Figure 4.119). If the cross-sectional area is A, determine the dimensions of the cross section which minimize the perimeter.

A landscape architect plans to enclose a 3000 square-foot rectangular region in a botanical garden. She will use shrubs costing $45 per foot along three sides and fencing costing $20 per foot along the fourth side. Find the minimum total cost.

A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions.

A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions.

A manufacturers cost of producing a product is given in Figure 4.120. The manufacturer can sell the product for a price p each (regardless of the quantity sold), so that the total revenue from selling a quantity q is R(q) = pq. (a) The difference (q) = R(q) C(q) is the total profit. For which quantity q0 is the profit a maximum? Mark your answer on a sketch of the graph. (b) What is the relationship between p and C (q0)? Explain your result both graphically and analytically. What does this mean in terms of economics? (Note that p is the slope of the line R(q) = pq. Note also that (q) has a maximum at q = q0, so  (q0)=0.) (c) Graph C (q) and p (as a horizontal line) on the same axes. Mark q0 on the q-axis

Using the cost and revenue graphs in Figure 4.120, sketch the following functions. Label the points q1 and q2. (a) Total profit (b) Marginal cost (c) Marginal revenue

A ship is steaming due north at 12 knots (1 knot = 1.85 kilometers/hour) and sights a large tanker 3 kilometers away northwest steaming at 15 knots due east. For reasons of safety, the ships want to maintain a distance of at least 100 meters between them. Use a calculator or computer to determine the shortest distance between them if they remain on their current headings, and hence decide if they need to change course

A polystyrene cup is in the shape of a frustum (the part of a cone between two parallel planes cutting the cone), has top radius 2r, base radius r and height h. The surface area S of such a cup is given by S = 3rr2 + h2 and its volume V by V = 7r2h/3. If the cup is to hold 200 ml, use a calculator or a computer to estimate the value of r that minimizes its surface area.

Suppose g(t) = (ln t)/t for t > 0. (a) Does g have either a global maximum or a global minimum on 0 <t< ? If so, where, and what are their values? (b) What does your answer to part (a) tell you about the number of solutions to the equation ln x x = ln 5 5 ?

For a > 0, the following line forms a triangle in the first quadrant with the x- and y-axes: a(a2 + 1)y = a x. (a) In terms of a, find the x- and y-intercepts of the line. (b) Find the area of the triangle, as a function of a. (c) Find the value of a making the area a maximum. (d) What is this greatest area? (e) If you want the triangle to have area 1/5, what choices do you have for a?

(a) Water is flowing at a constant rate (i.e., constant volume per unit time) into a cylindrical container standing vertically. Sketch a graph showing the depth of water against time. (b) Water is flowing at a constant rate into a cone-shaped container standing on its point. Sketch a graph showing the depth of the water against time.

The vase in Figure 4.121 is filled with water at a constant rate (i.e., constant volume per unit time). (a) Graph y = f(t), the depth of the water, against time, t. Show on your graph the points at which the concavity changes. (b) At what depth is y = f(t) growing most quickly? Most slowly? Estimate the ratio between the growth rates at these two depths.

A chemical storage tank is in the shape of an inverted cone with depth 12 meters and top radius 5 meters. When the depth of the chemical in the tank is 1 meter, the level is falling at 0.1 meters per minute. How fast is the volume of chemical changing?

In Problems 8283, describe the form of the limit (0/0, /, 0, , 1, 00, 0, or none of these). Does lHopitals rule apply? If so, explain how

In Problems 8283, describe the form of the limit (0/0, /, 0, , 1, 00, 0, or none of these). Does lHopitals rule apply? If so, explain how

In Problems 8487, determine whether the limit exists, and where possible evaluate it.

In Problems 8487, determine whether the limit exists, and where possible evaluate it.

In Problems 8487, determine whether the limit exists, and where possible evaluate it.

In Problems 8487, determine whether the limit exists, and where possible evaluate it.

The rate of change of a population depends on the current population, P, and is given by dP dt = kP(L P) for positive constants k, L. (a) For what nonnegative values of P is the population increasing? Decreasing? For what values of P does the population remain constant? (b) Find d2P/dt2 as a function of P

A spherical cell is growing at a constant rate of 400 m3/day (1 m= 106 m). At what rate is its radius increasing when the radius is 10 m?

A raindrop is a perfect sphere with radius r cm and surface area S cm2. Condensation accumulates on the raindrop at a rate equal to kS, where k = 2 cm/sec. Show that the radius of the raindrop increases at a constant rate and find that rate

A horizontal disk of radius a centered at the origin in the xy-plane is rotating about a vertical axis through the center. The angle between the positive x-axis and a radial line painted on the disk is radians. (a) What does d/dt represent? (b) What is the relationship between d/dt and the speed v of a point on the rim?

The depth of soot deposited from a smokestack is given by D = K(r + 1)er, where r is the distance from the smokestack. What is the relationship between the rate of change of r with respect to time and the rate of change of D with respect to time?

The mass of a circular oil slick of radius r is M = K (r ln(1 + r)), where K is a positive constant. What is the relationship between the rate of change of the radius with respect to t and the rate of change of the mass with respect to tim

Ice is being formed in the shape of a circular cylinder with inner radius 1 cm and height 3 cm. The outer radius of the ice is increasing at 0.03 cm per hour when the outer radius is 1.5 cm. How fast is the volume of the ice increasing at this time

Sand falls from a hopper at a rate of 0.1 cubic meters per hour and forms a conical pile beneath. If the side of the cone makes an angle of /6 radians with the vertical, find the rate at which the height of the cone increases. At what rate does the radius of the base increase? Give both answers in terms of h, the height of the pile in meters.

(a) A hemispherical bowl of radius 10 cm contains water to a depth of h cm. Find the radius of the surface of the water as a function of h. (b) The water level drops at a rate of 0.1 cm per hour. At what rate is the radius of the water decreasing when the depth is 5 cm?

A particle lies on a line perpendicular to a thin circular ring and through its center. The radius of the ring is a, and the distance from the point to the center of the ring is y. For a positive constant K, the gravitational force exerted by the ring on the particle is given by F = Ky (a2 + y2)3/2 , If y is increasing, how is F changing with respect to time, t, when (a) y = 0 (b) y = a/2 (c) y = 2a

A voltage, V volts, applied to a resistor of R ohms produces an electric current of I amps where V = IR. As the current flows the resistor heats up and its resistance falls. If 100 volts is applied to a resistor of 1000 ohms the current is initially 0.1 amps but rises by 0.001 amps/minute. At what rate is the resistance falling if the voltage remains constant?

A train is heading due west from St. Louis. At noon, a plane flying horizontally due north at a fixed altitude of 4 miles passes directly over the train. When the train has traveled another mile, it is going 80 mph, and the plane has traveled another 5 miles and is going 500 mph. At that moment, how fast is the distance between the train and the plane increasing

A fixed quantity of gas is allowed to expand at constant temperature. Find the rate of change of pressure with respect to volume

A fixed quantity of gas is allowed to expand at constant temperature. Find the rate of change of pressure with respect to volume

A population, P, in a restricted environment may grow with time, t, according to the logistic function P = L 1 + Cekt where L is called the carrying capacity and L, C and k are positive constants. (a) Find lim t P. Explain why L is called the carrying capacity. (b) Using a computer algebra system, show that the graph of P has an inflection point at P = L/2.

For positive a, consider the family of functions y = arctan x + a 1 ax , x> 0. (a) Graph several curves in the family and describe how the graph changes as a varies. (b) Use a computer algebra system to find dy/dx, and graph the derivative for several values of a. What do you notice? (c) Do your observations in part (b) agree with the answer to part (a)? Explain. [Hint: Use the fact that ax = a x for a > 0, x > 0.]

The function arcsinh x is the inverse function of sinh x. (a) Use a computer algebra system to find a formula for the derivative of arcsinh x. (b) Derive the formula by hand by differentiating both sides of the equation sinh(arcsinh x) = x. [Hint: Use the identity cosh2 x sinh2 x = 1.]

The function arccosh x, for x 0, is the inverse function of cosh x, for x 0. (a) Use a computer algebra system to find the derivative of arccosh x. (b) Derive the formula by hand by differentiating both sides of the equation cosh(arccosh x) = x, x 1. [Hint: Use the identity cosh2 x sinh2 x = 1.]

Consider the family of functions f(x) = a + x a + x, x 0, for positive a. (a) Using a computer algebra system, find the local maxima and minima of f.

a) Use a computer algebra system to find the derivative of y = arctan 1 cos x 1 + cos x  . (b) Graph the derivative. Does the graph agree with the answer you got in part (a)? Explain using the identity cos(x) = cos2(x/2) sin2(x/2).

In 1696, the first calculus textbook was published by the Marquis de lHopital. The following problem is a simplified version of a problem from this text. In Figure 4.122, two ropes are attached to the ceiling at points 3 meters apart. The rope on the left is 1 meter long and has a pulley attached at its end. The rope on the right is 3 meters long; it passes through the pulley and has a weight tied to its end. When the weight is released, the ropes and pulley arrange themselves so that the distance from the weight to the ceiling is maximized. (a) Show that the maximum distance occurs when the weight is exactly halfway between the the points where the ropes are attached to the ceiling. [Hint: Write the vertical distance from the weight to the ceiling in terms of its horizontal distance to the point at which the left rope is tied to the ceiling. A computer algebra system will be useful.] (b) Does the weight always end up halfway between the ceiling anchor points no matter how long the lefthand rope is? Explain.