A1. A function f is continuous, differentiable, and strictly decreasing on the interval [2.5,5]; some values of f are shown in the table above.

(a) Estimate f′(4.0) and f′(4.8).
(b) What does the table suggest may be true of the concavity of f? Explain.
(c) Estimate  with a Riemann Sum using left endpoints.
(d) Set up (but do not evaluate) a Riemann Sum that estimates the volume of the solid formed when f is rotated around the x-axis.

A2. An equation of the tangent line to the curve x²y – x = y³ – 8 at the point (0,2) is 12y + x = 24.

(a) Given that the point (0.3,y₀) is on the curve, find y₀ approximately, using the tangent line.
(b) Find the true value of y₀.
(c) What can you conclude about the curve near x = 0 from your answers to parts (a) and (b)?

A3. A differentiable function f defined on –7 < x < 7 has f(0) = 0 and . (Note: The following questions refer to f, not to f′.)

(a) Describe the symmetry of f.
(b) On what intervals is f decreasing?
(c) For what values of x does f have a relative maximum? Justify your answer.
(d) How many points of inflection does f have? Justify your answer.

A4. Let C represent the piece of the curve  that lies in the first quadrant. Let S be the region bounded by C and the coordinate axes.

(a) Find the slope of the line tangent to C at y = 1.
(b) Find the area of S.
(c) Find the volume generated when S is rotated about the x-axis.

A5. Let R be the point on the curve of y = x – x² such that the line OR (where O is the origin) divides the area bounded by the curve and the x-axis into two regions of equal area. Set up (but do not solve) an integral to find the x-coordinate of R.

A6. Suppose f″ = sin (2^x) for –1 < x < 3.2.

(a) On what intervals is the graph of f concave downward? Justify your answer.
(b) Find the x-coordinates of all relative minima of f′.
(c) How many points of inflection does the graph of f′ have? Justify your answer.

A7. Let f(x) = cos x and g(x) = x² – 1.

(a) Find the coordinates of any points of intersection of f and g.
(b) Find the area bounded by f and g.

A8. (a) In order to investigate mail-handling efficiency, at various times one morning a local post office checked the rate (letters/min) at which an employee was sorting mail. Use the results shown in the table and the trapezoid method to estimate the total number of letters he may have sorted that morning.

(b) Hoping to speed things up a bit, the post office tested a sorting machine that can process mail at the constant rate of 20 letters per minute. The graph shows the rate at which letters arrived at the post office and were dumped into this sorter.

(i) When did letters start to pile up?
(ii) When was the pile the biggest?
(iii) How big was it then?
(iv) At about what time did the pile vanish?

A9. Let R represent the region bounded by y = sin x and y = x⁴. Find:

(a) the area of R
(b) the volume of the solid whose base is R if all cross sections perpendicular to the x-axis are isosceles triangles with height 3
(c) the volume of the solid formed when R is rotated around the x-axis

A10. The town of East Newton has a water tower whose tank is an ellipsoid, formed by rotating an ellipse about its minor axis. Since the tank is 20 feet tall and 50 feet wide, the equation of the ellipse is .

(a) If there are 7.48 gallons of water per cubic foot, what is the capacity of this tank to the nearest thousand gallons?
(b) East Newton imposes water rationing whenever the tank is only one-quarter full. Write an equation to find the depth of the water in the tank when rationing becomes necessary. (Do not solve.)

A11. 

Note: Scales are different on the three figures.

The sides of the watering trough above are made by folding a sheet of metal 24 inches wide and 5 feet (60 inches) long at an angle of 60°, as shown above. Ends are added, and then the trough is filled with water.

(a) If water pours into the trough at the rate of 600 cubic inches per minute, how fast is the water level rising when the water is 4 inches deep?
(b) Suppose, instead, the sheet of metal is folded twice, keeping the sides of equal height and inclined at an angle of 60°, as shown. Where should the folds be in order to maximize the volume of the trough? Justify your answer.

A12. Given the function f(x) = e^2x(x² – 2):

(a) For what values of x is f decreasing?
(b) Does this decreasing arc reach a local or a global minimum? Justify your answer.
(c) Does f have a global maximum? Justify your answer.

A13. (a) A spherical snowball melts so that its surface area shrinks at the constant rate of 10 square centimeters per minute. What is the rate of change of volume when the snowball is 12 centimeters in diameter?

(b) The snowball is packed most densely nearest the center. Suppose that, when it is 12 centimeters in diameter, its density x centimeters from the center is given by  grams per cubic centimeter. Set up an integral for the total number of grams (mass) of the snowball then. Do not evaluate.

*A14. (a) Using your calculator, verify that

(b) Use the Taylor polynomial of degree 7 about 0:

to approximate tan^–11/5 and the polynomial of degree 1 to approximate tan^–1 1/239.
(c) Use part (b) to evaluate the expression in (a).
(d) Explain how the approximation for π/4 given here compares with that obtained using π/4 = tan^–1 1.

*A15. (a) Show that the series  converges.

(b) How many terms of the series are needed to get a partial sum within 0.1 of the sum of the whole series?
(c) Tell whether the series  is absolutely convergent, conditionally convergent, or divergent. Justify your answer.

*A16. Given  with y = 2 at t = 0 and y = 5 at t = 2:

(a) Find k.
(b) Express y as a function of t.
(c) For what value of t will y = 8?
(d) Describe the long-range behavior of y

*A17. An object P is in motion in the first quadrant along the parabola y = 18 – 2x² in such a way that at t seconds the x-value of its position is .

(a) Where is P when t = 4?
(b) What is the vertical component of its velocity there?
(c) At what rate is its distance from the origin changing at t = 4?
(d) When does it hit the x-axis?
(e) How far did it travel altogether?

*A18. A particle moves in the xy-plane in such a way that at any time t ≥ 0 its position is given by x(t) = 4 arctan t, .

(a) Sketch the path of the particle, indicating the direction of motion.
(b) At what time t does the particle reach its highest point? Justify.
(c) Find the coordinates of that highest point, and sketch the velocity vector there.
(d) Describe the long-term behavior of the particle.

*A19. Let R be the region bounded by the curve r = 2 + cos 2θ, as shown.

(a) Find the dimensions of the smallest rectangle that contains R and has sides parallel to the x- and y-axes.
(b) Find the area of R.

*A20. 

(a) For what positive values of x does  converge?
(b) How many terms are needed to estimate f(0.5) to within 0.01?
(c) Would an estimate for f(–0.5) using the same number of terms be more accurate, less accurate, or the same? Explain.

*A21. After pollution-abatement efforts, conservation researchers introduce 100 trout into a small lake. The researchers predict that after m months the population, F, of the trout will be modeled by the differential equation .

(a) How large is the trout population when it is growing the fastest?
(b) Solve the differential equation, expressing F as a function of m.
(c) How long after the lake was stocked will the population be growing the fastest?

B1. Draw a graph of y = f(x), given that f satisfies all the following conditions:

(a) f′(–1) = f′(1) = 0.
(b) If x < –1, f′(x) > 0 but f″ < 0.
(c) If –1 < x < 0, f′(x) > 0 and f″ > 0.
(d) If 0 < x < 1, f′(x) > 0 but f″ < 0.
(E) If x > 1, f′(x) < 0 and f″ < 0.

B2. The figure below shows the graph of f′, the derivative of f, with domain –3 ≤ x ≤ 9. The graph of f′ has horizontal tangents at x = 2 and x = 4 and a corner at x = 6.

(a) Is f continuous? Explain.
(b) Find all values of x at which f attains a local minimum. Justify.
(c) Find all values of x at which f attains a local maximum. Justify.
(d) At what value of x does f attain its absolute maximum? Justify.
(e) Find all values of x at which the graph of f has a point of inflection. Justify.

B3. Find the area of the largest rectangle (with sides parallel to the coordinate axes) that can be inscribed in the region bounded by the graphs of f(x) = 8 – 2x² and g(x) = x² – 4.

B4. Given the graph of f(x), sketch the graph of f′(x).

B5. A cube is contracting so that its surface area decreases at the constant rate of 72 in.²/sec. Determine how fast the volume is changing at the instant when the surface area is 54 ft².

B6. A square is inscribed in a circle of radius a as shown in the diagram below. Find the volume obtained if the region outside the square but inside the circle is rotated about a diagonal of the square.

B7. 

(a) Sketch the region in the first quadrant bounded above by the line y = x + 4, below by the line y = 4 – x, and to the right by the parabola y = x² + 2.
(b) Find the area of this region.

B8. The graph shown below is based roughly on data from the U.S. Department of Agriculture.

(a) During which intervals did food production decrease in South Asia?
(b) During which intervals did the rate of change of food production increase?
(c) During which intervals did the increase in food production accelerate?

B9. A particle moves along a straight line so that its acceleration at any time t is given in terms of its velocity v by a = –2v.

(a) Find v in terms of t if v = 20 when t = 0.
(b) Find the distance the particle travels while v changes from v = 20 to v = 5.

B10. Let R represent the region bounded above by the parabola y = 27 – x² and below by the x-axis. Isosceles triangle AOB is inscribed in region R with its vertex at the origin O and its base  parallel to the x-axis. Find the maximum possible area for such a triangle.

B11. Newton’s law of cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and its surroundings.

It is 9:00 P.M., time for your milk and cookies. The room temperature is 68° when you pour yourself a glass of 40° milk and start looking for the cookie jar. By 9:03 the milk has warmed to 43°, and the phone rings. It’s your friend, with a fascinating calculus problem. Distracted by the conversation, you forget about the glass of milk. If you dislike milk warmer than 60°, how long, to the nearest minute, do you have to solve the calculus problem and still enjoy acceptably cold milk with your cookies?

B12. Let h be a function that is even and continuous on the closed interval [–4,4]. The function h and its derivatives have the properties indicated in the table below. Use this information to sketch a possible graph of h on [–4,4].

*B13. 

(a) Find the Maclaurin series for f(x) = ln (1 + x).
(b) What is the radius of convergence of the series in (a)?
(c) Use the first five terms in (a) to approximate ln (1.2).
(d) Estimate the error in (c), justifying your answer.

*B14. A cycloid is given parametrically by x = θ – sin θ, y = 1 – cos θ.

(a) Find the slope of the curve at the point where .
(b) Find an equation of the tangent to the cycloid at the point where .

*B15. Find the area of the region enclosed by both the polar curves r = 4 sin θ and r = 4 cos θ.

*B16.

(a) Find the 4th-degree Taylor polynomial about 0 for cos x.
(b) Use part (a) to evaluate 
(c) Estimate the error in (b), justifying your answer.

*B17. A particle moves on the curve of y³ = 2x + 1 so that its distance from the x-axis is increasing at the constant rate of 2 units/sec. When t = 0, the particle is at (0,1).

(a) Find a pair of parametric equations x = x(t) and y = y(t) that describe the motion of the particle for nonnegative t.
(b) Find |a|, the magnitude of the particle’s acceleration, when t = 1.

*B18. Find the area of the region that the polar curves r = 2 – cos θ and r = 3 cos θ enclose in common.