Investigate the family of curves given by the parametric equations x= t3 - ct, y =t2. In particular, determine the values of for which there is a loop and find the point where the curve intersects itself. What happens to the loop as c increases? Find the coordinates of the leftmost and rightmost points of the loop.
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = x² –
> (a). Investigate the family of polynomials given by the equation f (x) = 2x3 + cx2 + 2x. For what values of does the curve have maximum and minimum points? (b). Show that the minimum and maximum points of every curve in the family lie on the curve y = x
> (a). Investigate the family of polynomials given by the equation f (x) = cx4 – 2x2 + 1. For what values of does the curve have minimum points? (b). Show that the minimum and maximum points of every curve in the family lie on the parabola y = 1 – x2. Illu
> Investigate the family of curves given by the equation f (x) = x4 + cx2 + x. Start by determining the transitional value of at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> If f' is the function considered in Example 3, use a computer algebra system to calculate f' and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate f" and use it to estimate the intervals of concavity
> Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum
> Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. 2 x 10 f(x - x*
> Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line l parallel to the line joining the light sources and at a distance meter from it (see the figure). We want to locate P on l so that the intens
> The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite cha
> Investigate the family of curves given by f (x) = xne-x, where n is a positive integer. What features do these curves have in common? How do they differ from one another? In particular, what happens to the maximum and minimum points and inflection points
> Find the dimensions of a rectangle with area whose perimeter is as small as possible.
> Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls
> (a). Graph the function. (b). Explain the shape of the graph by computing the limit as x→0 or as x→∞. (c). Estimate the maximum and minimum values and then use calculus to find the exact values. (d). Use a graph of f" to estimate the x-coordinates of the
> (a). Graph the function. (b). Explain the shape of the graph by computing the limit as x→0 or as x→∞. (c). Estimate the maximum and minimum values and then use calculus to find the exact values. (d). Use a graph of f" to estimate the x-coordinates of the
> (a). Graph the function. (b). Use l’Hospital’s Rule to explain the behavior as x→0. (c). Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. 1/1 f(x
> Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W. [Hint: Express the area as a function of an angle θ.]
> A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? 6 -- -
> Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. 8 f(x) = 1 +-+
> Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when &Ic
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = tan x
> The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? b a a
> Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
> Let and be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a, b).
> The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent sh
> A manufacturer has been selling 1000 television sets a week at $450 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of sets sold will increase by 100 per week. (a). Find the demand function. (b). How large a reba
> During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for @10 each and his sales averaged 20 per day. When he increased the price by $1, he found that the average decreased by two sales per day. (a). Fin
> A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a). Find the demand function, assuming that
> (a). Show that if the profit P (x) is a maximum, then the marginal revenue equals the marginal cost. (b). If C (x) 16,000 + 500x – 1.6x2 + 0.004x3 is the cost function and p (x) = 1700 – 7x is the demand function, find the production level that will maxi
> (a). If C (x) is the cost of producing units of a commodity, then the average cost per unit is c (x) = C (x)/x. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b). If C (x) = 16,000 + 200x + 4x3/2, in dollars,
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = r - x
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. x² – 1 f(x):
> At which points on the curve y = 1 + 40x3 – 3x5 does the tangent line have the largest slope?
> The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?
> Find an equation of the line through the point (3, 5) that cuts off the least area from the first quadrant.
> A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mi/h and row a boat
> Given that which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. lim f(x) = 0 lim g(x) = 0 lim h(x) = 1 %3D エ→ エ→ロ lim p(x) = 0 lim q(x) = 0 %3D f(x) (a) lim g(x) 1-a
> Suppose the refinery in Exercise 35 is located 1 km north of the river. Where should P be located? Exercise 35: An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to sto
> An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000/k
> A boat leaves a dock at 2:00 PM and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 PM. At what time were the two boats closest together?
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> The graph shows the fuel consumption of a car (measured in gallons per hour) as a function of the speed of the car. At very low speeds the engine runs inefficiently, so initially decreases as the speed increases. But at high speeds the fuel consumption i
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = x – 1
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> The family of functions f (t) = C (e-at – e-bt), where a, b, and C are positive numbers and b > a, has been used to model the concentration of a drug injected into the bloodstream at time t = 0. Graph several members of this family. What do they have in
> Graph the curve in a viewing rectangle that displays all the important aspects of the curve. At what points does the curve have vertical or horizontal tangents? x = t* + 41 – 81?, y= 2t2 – t
> Graph the curve in a viewing rectangle that displays all the important aspects of the curve. At what points does the curve have vertical or horizontal tangents? x = t* – 2t° – 2t', y=r°- t
> Use a graph to estimate the coordinates of the leftmost point on the curve x = t4 – t2, y = t + ln t. Then use calculus to find the exact coordinates.
> In Example 4 we considered a member of the family of functions f (x) = sin (x + sin cs) that occur in FM synthesis. Here we investigate the function with c = 3. Start by graphing f in the viewing rectangle [0, π] by [-1.2, 1.2]. How many local maximum po
> Graph f (x) = ex + ln |x – 4| using as many viewing rectangles as you need to depict the true nature of the function.
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = x° –
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. 1- elz 1+ el f(x) = %3D
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. f(x) = (x² – 1)e' arctan x
> In this project we investigate the most economical shape for a can. We first interpret this to mean that the volume V of a cylindrical can is given and we need to find the height and radius that minimize the cost of the metal to make the can (see the fig
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> If f' is continuous, f (2) = 0, and f'(2) = 7, evaluate limx→0 f (2 + 3x) + f (2 + 5x)/x
> Suppose is a positive function. If limx→a f (x) = 0 and limx→a g (x) = ∞, show that limx→a [f (x)] g (x) = 0. This shows that is not an indeterminate form.
> Evaluate limx→∞ [x - x2 ln (1 + x/x)].
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. f(x) = x+ 5 sin x, x< 20
> The first appearance in print of l’Hospital’s Rule was in the book Analyse des Infiniment Petits published by the Marquis de l’Hospital in 1696. This was the first calculus textbook ever published and
> A metal cable has radius and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is where c is a positive constant. Find the following
> If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is Show that lim E→0+ P (E) = 0. e + e-E P(E) eE – e-E E
> If an object with mass m is dropped from rest, one model for its speed after seconds, taking air resistance into account, is where g is the acceleration due to gravity and is a positive constant. (In Chapter 7 we will be able to deduce this equation from
> If an initial amount A0 of money is invested at an interest rate r compounded times a year, the value of the investment after t years is If we let n→∞, we refer to the continuous compounding of interest. Use lâ
> Prove that for any number p > 0. This shows that the logarithmic function approaches ∞ more slowly than any power of x. In x = 0 lim rP
> Prove that for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x. lim = 00
> Investigate the family of curves f (x) = ex - cx. In particular, find the limits as x→±∞ and determine the values of for which has an absolute minimum. What happens to the minimum points as increases?
> What happens if you try to use l’Hospital’s Rule to evaluate Evaluate the limit using another method. lim x² + 1
> The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. 2/3 f(x) 1+x + x*
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> A painting in an art gallery has height and is hung so that its lower edge is a distance above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer sta
> Where should the point P be chosen on the line segment AB so as to maximize the angle θ? В 2 P 3 A
> A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water? -
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. f(x) = x? +x + 1
> Use l’Hospital’s Rule to help find the asymptotes of f. Then use them, together with information from f' and f", to sketch the graph of f. Check your work with a graphing device. f(x) = xe
> The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y? 12 - y 8
> Use l’Hospital’s Rule to help find the asymptotes of f. Then use them, together with information from f' and f", to sketch the graph of f. Check your work with a graphing device. f(x) = e"/x
> Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. S
> Illustrate l’Hospital’s Rule by graphing both f (x)/g (x) and f'(x)/ g'(x) near x = 0 to see that these ratios have the same limit as x→0. Also, calculate the exact value of the limit. f(x) = 2x
> Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
> Illustrate l’Hospital’s Rule by graphing both f (x)/g (x) and f'(x)/ g'(x) near x = 0 to see that these ratios have the same limit as x→0. Also, calculate the exact value of the limit. f(x) = e*
> Use a graph to estimate the value of the limit. Then use l’Hospital’s Rule to find the exact value. 5 - 4" lim 1-0 3* - 2*
> Use a graph to estimate the value of the limit. Then use l’Hospital’s Rule to find the exact value.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y = 4 – x2 at some point?
> What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3/x at some point?
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> If is the function of Exercise 12, find f' and f" and use their graphs to estimate the intervals of increase and decrease and concavity of f. Exercise 12: Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sket
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 f
