If a metal ball with mass m is projected in water and the force of resistance is

Explain the difference between an absolute maximum and a local maximum. Illustrate with a sketch.

(a) What does the Extreme Value Theorem say? (b) Explain how the Closed Interval Method works.

(a) State Fermats Theorem. (b) Define a critical number of f.

(a) State Rolles Theorem. (b) State the Mean Value Theorem and give a geometric interpretation.

(a) State the Increasing/Decreasing Test. (b) What does it mean to say that f is concave upward on an interval I? (c) State the Concavity Test. (d) What are inflection points? How do you find them?

(a) State the First Derivative Test. (b) State the Second Derivative Test. (c) What are the relative advantages and disadvantages of these tests?

(a) What does lHospitals Rule say? (b) How can you use lHospitals Rule if you have a product fsxd tsxd where fsxd l 0 and tsxd l ` as x l a? (c) How can you use lHospitals Rule if you have a difference fsxd 2 tsxd where fsxd l ` and tsxd l ` as x l a? (d) How can you use lHospitals Rule if you have a power f fsxdgtsxd where fsxd l 0 and tsxd l 0 as x l a?

State whether each of the following limit forms is indeterminate. Where possible, state the limit. (a) 0 0 (b) ` ` (c) 0 ` (d) ` 0 (e) ` 1 ` (f) ` 2 ` (g) ` ? ` (h) ` ? 0 (i) 00 (j) 0` (k) `0 (l) 1`

If you have a graphing calculator or computer, why do you need calculus to graph a function?

(a) Given an initial approximation x1 to a root of the equation fsxd 0, explain geometrically, with a diagram, how the second approximation x2 in Newtons method is obtained. (b) Write an expression for x2 in terms of x1, fsx1d, and f9sx1d. (c) Write an expression for xn11 in terms of xn, fsxn d, and f9sxnd. (d) Under what circumstances is Newtons method likely to fail or to work very slowly?

(a) What is an antiderivative of a function f ? (b) Suppose F1 and F2 are both antiderivatives of f on an interval I. How are F1 and F2 related?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and t are increasing on an interval I, then f 2 t is increasing on I.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and t are increasing on an interval I, then f t is increasing on I.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and t are positive increasing functions on an interval I, then f t is increasing on I.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is increasing and fsxd . 0 on I, then tsxd 1yfsxd is decreasing on I.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is even, then f9 is even.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. . If f is periodic, then f9 is periodic.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The most general antiderivative of fsxd x 22 is Fsxd 2 1 x 1 C

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f9sxd exists and is nonzero for all x, then fs1d fs0d.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If lim xl` fsxd 1 and limxl` tsxd `, then lim xl` f fsxdg tsxd 1

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. lim xl0 x ex 1

Use the guidelines of Section 4.5 to sketch the curve. y x 1 2 x 2

Use the guidelines of Section 4.5 to sketch the curve. y 1 xsx 2 3d 2

Use the guidelines of Section 4.5 to sketch the curve. y 1 x 2 2 1 sx 2 2d 2

Use the guidelines of Section 4.5 to sketch the curve. y sx 2 1d 3 x 2

Use the guidelines of Section 4.5 to sketch the curve.y s1 2 x 1 s1 1 x

Use the guidelines of Section 4.5 to sketch the curve. y xs2 1 x

Use the guidelines of Section 4.5 to sketch the curve. y x 2y3 sx 2 3d 2

Use the guidelines of Section 4.5 to sketch the curve.y ex sin x, 2 < x <

Use the guidelines of Section 4.5 to sketch the curve.y 4x 2 tan x, 2y2 , x , y2

Use the guidelines of Section 4.5 to sketch the curve. y sin21 s1yxd

Use the guidelines of Section 4.5 to sketch the curve. y e2x2x 2

Use the guidelines of Section 4.5 to sketch the curve.y sx 2 2de2x

Use the guidelines of Section 4.5 to sketch the curve.y x 1 lnsx 2 1 1d

Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly fsxd x 2 2 1 x 3

Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly fsxd x 3 1 1 x 6 1 1

Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly fsxd 3x 6 2 5x 5 1 x 4 2 5x 3 2 2x 2 1 2

Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly . fsxd x 2 1 6.5 sin x, 25 < x < 5

Graph f sxd e 21yx2 in a viewing rectangle that shows all the main aspects of this function. Estimate the inflection points. Then use calculus to find them exactly.

(a) Graph the function fsxd 1ys1 1 e 1yx d. (b) Explain the shape of the graph by computing the limits of fsxd as x approaches `, 2`, 01, and 02. (c) Use the graph of f to estimate the coordinates of the inflection points. (d) Use your CAS to compute and graph f 0. (e) Use the graph in part (d) to estimate the inflection points more accurately.

Use the graphs of f, f9, and f 0 to estimate the x-coordinates of the maximum and minimum points and inflection points of f. fsxd cos2 x sx 2 1 x 1 1 , 2 < x <

Use the graphs of f, f9, and f 0 to estimate the x-coordinates of the maximum and minimum points and inflection points of f. fsxd e20.1x lnsx 2 2 1d

Investigate the family of functions fsxd lnssin x 1 Cd. What features do the members of this family have in common? How do they differ? For which values of C is f continuous on s2`, `d? For which values of C does f have no graph at all? What happens as C l `?

Investigate the family of functions fsxd cxe2cx 2 . What happens to the maximum and minimum points and the inflection points as c changes? Illustrate your conclusions by graphing several members of the family.

Show that the equation 3x 1 2 cos x 1 5 0 has exactly one real root.

Suppose that f is continuous on f0, 4g, fs0d 1, and 2 < f9sxd < 5 for all x in s0, 4d. Show that 9 < fs4d < 21.

By applying the Mean Value Theorem to the function fsxd x 1y5 on the interval f32, 33g, show that 2 , s 5 33 , 2.0125

For what values of the constants a and b is s1, 3d a point of inflection of the curve y ax 3 1 bx 2 ?

Let tsxd fsx 2 d, where f is twice differentiable for all x, f9sxd . 0 for all x 0, and f is concave downward on s2`, 0d and concave upward on s0, `d. (a) At what numbers does t have an extreme value? (b) Discuss the concavity of t.

Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.

Show that the shortest distance from the point sx1, y1d to the straight line Ax 1 By 1 C 0 is | Ax1 1 By1 1 C | sA2 1 B2

Find the point on the hyperbola xy 8 that is closest to the point s3, 0d.

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r

Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.

In DABC, D lies on AB, CD  AB, | AD | | BD | 4 cm, and | CD | 5 cm. Where should a point P be chosen on CD so that the sum | PA | 1 | PB | 1 | PC | is a minimum?

Solve Exercise 55 when | CD | 2 cm

The velocity of a wave of length L in deep water is v K L C 1 C L where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?

A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?

A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?

A manufacturer determines that the cost of making x units of a commodity is Csxd 1800 1 25x 2 0.2x 2 1 0.001x 3 and the demand function is psxd 48.2 2 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost.

Use Newtons method to find the root of the equation x5 2 x4 1 3x2 2 3x 2 2 0 in the interval f1, 2g correct to six decimal places.

Use Newtons method to find all solutions of the equation sin x x 2 2 3x 1 1 correct to six decimal places.

Use Newtons method to find the absolute maximum value of the function fstd cos t 1 t 2 t 2 correct to eight decimal places.

Use the guidelines in Section 4.5 to sketch the curve y x sin x, 0 < x < 2. Use Newtons method when necessary

Find the most general antiderivative of the function. fsxd 4sx 2 6x 2 1 3

Find the most general antiderivative of the function. tsxd 1 x 1 1 x 2 1 1

Find the most general antiderivative of the function. fstd 2 sin t 2 3e

Find the most general antiderivative of the function. fsxd x23 1 cosh x

Find f. f9std 2t 2 3 sin t, fs0d 5

Find f. f9sud u2 1 su u , fs1d 3

Find f. f 0sxd 1 2 6x 1 48x 2 , fs0d 1, f9s0d 2

Find f. f 0sxd 5x 3 1 6x 2 1 2, fs0d 3, fs1d 22

A particle is moving with the given data. Find the position of the particle. vstd 2t 2 1ys1 1 t 2 d, ss0d 1

A particle is moving with the given data. Find the position of the particle. astd sin t 1 3 cos t, ss0d 0, vs0d 2

(a) If fsxd 0.1ex 1 sin x, 24 < x < 4, use a graph of f to sketch a rough graph of the antiderivative F of f that satisfies Fs0d 0. (b) Find an expression for Fsxd. (c) Graph F using the expression in part (b). Compare with your sketch in part (a).

Investigate the family of curves given by fsxd x 4 1 x 3 1 cx 2 In particular you should determine the transitional value of c at which the number of critical numbers changes and the transitional value at which the number of inflection points changes. Illustrate the various possible shapes with graphs.

A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 mys. Will it burst?

In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.

A rectangular beam will be cut from a cylindrical log of radius 10 inches. (a) Show that the beam of maximal cross-sectional area is a square. (b) Four rectangular planks will be cut from the four sections of the log that remain after cutting the square beam. Determine the dimensions of the planks that will have maximal cross-sectional area. (c) Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.

If a projectile is fired with an initial velocity v at an angle of inclination  from the horizontal, then its trajectory, neglecting air resistance, is the parabola y stan dx 2 t 2v2 cos2  x 2 0 ,  ,  2 (a) Suppose the projectile is fired from the base of a plane that is inclined at an angle ,  . 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by Rsd 2v 2 cos  sins 2 d t cos2  (b) Determine  so that R is a maximum. (c) Suppose the plane is at an angle  below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R.

If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is PsEd e E 1 e2E e E 2 e2E 2 1 E Show that lim E l 01 PsEd 0.

If a metal ball with mass m is projected in water and the force of resistance is proportional to the square of the velocity, then the distance the ball travels in time t is sstd m c ln coshtc mt where c is a positive constant. Find lim cl 01 sstd.

Show that, for x . 0, x 1 1 x 2 , tan21 x , x

Sketch the graph of a function f such that f9sxd , 0 for all x, f 0sxd . 0 for | x | . 1, f 0sxd , 0 for | x | , 1, and limx l6` f fsxd 1 xg 0.

A light is to be placed atop a pole of height h feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle  (see the figure) and inversely proportional to the square of the distance d from the source. (a) How tall should the light pole be to maximize I? (b) Suppose that the light pole is h feet tall and that a woman is walking away from the base of the pole at the rate of 4 ftys. At what rate is the intensity of the light at the point on her back 4 ft above the ground decreasing when she reaches the outer edge of the traffic circle?

Water is flowing at a constant rate into a spherical tank. Let Vstd be the volume of water in the tank and Hstd be the height of the water in the tank at time t. (a) What are the meanings of V9std and H9std? Are these derivatives positive, negative, or zero? (b) Is V 0std positive, negative, or zero? Explain. (c) Let t1, t2, and t3 be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values H0st1d, H0st2d, and H0st3d positive, negative, or zero? Why?