The italicized phrase before each sample problem indicates a similar problem in James Stewart. 2007. Essential Calculus: Early Transcendentals. Thomson Brooks/Cole: Belmont, CA.
Similar to 4.1.14:
4.1 Graphs identifying local and absolute extrema, and critical numbers.
a. Sketch a function with two relative (local) minima, one relative maximum, and no absolute maximum.
b. Sketch a function with four relative maxima, two relative minima, and eight critical numbers.
Similar to 4.1.44:
4.2 Absolute maxima and minima.
For the given interval of the function, determine the absolute maximum and absolute minimum.
f(x) = -3x / (-x^2 - 4), [-1,+4]
Similar to 4.2.16:
4.3 Mean Value Theorem hypothesis conditions.
Prove that f(1) - f(0) = f'(c)(1 - 0) is not true for any real c. Explain why this does not disprove the Mean Value Theorem.
f(x) = (x - 3) / (2x - 1)
Similar to 4.2.36:
4.4 Using derivatives to find graph shapes.
f(x) = 3x^3 / (2x + 1)^3
a. Determine vertical and horizontal asymptotes.
b. Determine where the function is increasing or decreasing.
c. Determine relative (local) maximum and minimum values.
d. Determine where the function is concave up or down, and the location of inflection points.
e. Sketch f based on what you determined above.
Similar to 4.3.48:
4.5 Consider the family of curves below:
y = a + {1/[b*sqrt(2pi)]}*exp[-(x-c)^2/(2b^2)], where a, b, and c are constants.
Let's eliminate the 1/[b*sqrt(2pi)] factor, and let a = 0.5, b = 1, and c = 2 to obtain
f(x) = 0.5 + exp{[-(x-2)^2]/2}
a. Determine the maximum value, asymptote, and inflection points of f.
b. How does the value of b affect the shape of the curve?
c. Show b's effect by graphing the equation below on the same screen for four different values of b.
g(x) = 0.5 + exp{[-(x-2)^2]/b}
Similar to 4.3.50:
4.6 Find the values for the constants a and b in the function f(x) below if f(2) = ln 3 and f ' (2) = (1/2) ln 3 + (1/2).
f(x) = ax ln (bx)
Similar to 4.5.2:
4.7 Find the sum of two numbers whose product is a minimum, and difference is 76.
Similar to 4.5.10:
4.8 A hollow cylinder of uniform thickness, with a circular base and completely open on the opposite end, is required to have a volume of 1000 pi cubic centimeters. Find the radius and height of such a cylinder that will require the least material to construct.
Similar to 4.5.32:
4.9 Romeo is a hungry bullfrog with a very good sense of smell. But he claims to have a terrible sense of direction. He is at the north point of a triangular island that's 80 feet on each side. And, as soon as possible, he wants to meet his mate Juliet at the exact middle of island's south shore.
Initially, Romeo can swim and hop 8 feet per second.
His swimming and hopping speeds are always equal, but, as he gets more to eat, both will increase uniformly with distance traveled. While swimming, he can catch enough food to increase his speed by another 10 percent every 10 feet.
Frequently in the past, Romeo has claimed chasing food on land was a very disorienting, very time-consuming distraction. But even on land, he retains any speed he gained from eating while swimming. (The percent gained will always be of his original speed before eating anything.)
And fortunately, the honeysuckle is blooming.
Honeysuckle vines grow exclusively in a narrow band about a line from the island's north point to the exact middle of the south shore. By strictly focusing on its smell, Romeo can hop to the nearest honeysuckle vine, and he can follow those vines south. Swimming, he can also follow the island's shore.
But, if Romeo hops in any direction on the island other than toward the nearest honeysuckle, he gets lost. (Juliet suspects he lets the tree frogs beguile him.)
By the quickest route, how long will it take Romeo to reach Juliet?
(For route segments with uniformly increasing speed, approximate the average speed as half the sum of the initial and final speeds.)
Similar to 4.5.40:
4.10 John, the owner of a pizza restaurant chain, has the raw materials to make 10,000 pizzas this month. From past experience, he estimates that if he charges $7.00 per pizza, he can sell 10,000 pizzas, but for each $0.50 increase in price he will sell 450 fewer pizzas. What should he charge per pizza if he wants to maximize his revenue?
Similar to 4.7.32:
4.11 f is graphed below. Which of the other three graphs is the antiderivative of f? How do justify your choice?
Similar to 4.7.48:
4.12 Tom Sawyer wants to get rid of his wart before Becky Thatcher returns from St. Louis. But he also wants to keep the apple he got from his Aunt Polly's fence needing whitewashing again. So rather than trade, he bets his apple against Huckleberry Finn's dead cat--the only wart cure, but a sure one--that the river is flowing faster than 5 feet per minute.
He and Huck agree to ask the next boatman with a watch to time the river's flow for them.
Meanwhile for a second time, standing 3 feet from his raft's leading end, Huck throws a rock tied to and wrapped in a parachute straight up into the breezeless air.
The rock leaves his hand (6 feet above the raft) at 96 feet per second. The parachute opens 3 1/2 seconds later and for next 1 1/2 seconds the rock's fall slows at a constant rate to 1 foot per second.
(Before the parachute opens, gravity slows the rock's climb 32 feet per second squared and then speeds its fall by the same rate.)
(a) Write expressions for the height, y, and the velocity, v, for all times t. Graph y and v.
(b) When does the rock reach its maximum height and what is that height?
(c) Suppose the length of Huck's 25 foot raft is parallel to the river's flow and the rock lands 3 1/2 feet from the raft's trailing end. Will Tom still have his wart next time he sees Becky?
