Whitman MATH 125 - Review Questions
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Review Questions, Exam 3Math 125, Fall 2006Exam 3 will cover material from 3.10-4.3. You should also look over the homework and quizzes.1. A child is flying a kite. If the kite is 90 feet above the child’s hand level and the windis blowing it on a horizontal course at 5 feet per second, how fast is the child payingout cord when 150 feet of cord is out? (Assume that the cord forms a line- actually anunrealistic assumption).2. Use differentials to approximate the increase in area of a soap bubble, when its radiusincreases from 3 inches to 3.025 inches (A = 4πr2)3. True or False, and give a short reason:(a) If air is being pumped into a spherical rubber balloon at a constant rate, then theradius will increase, but at a slower and slower rate.(b) If f(x), g(x) are increasing on an interval I, so is f(x)g(x).(c) If y = x5, then dy ≥ 0(d) If a car averages 60 miles per hour over an interval of time, then at some instant,the speedometer must have read exactly 60.(e) A global maximum is always a local maximum.(f) The linear function f(x) = ax + b, where a, b are constant, and a 6= 0, has nominimum value on any open interval. (An interval is open if it does not include itsendpoints).(g) Suppose P and Q are two points on the surface of the sea, with Q lying generallyto the east of P . It is possible to sail from P to Q (always sailing roughly east),without ever sailing in the exact direction from P to Q.(h) If f(x) = 0 has three (distinct) real solutions, then f0(x) = 0 must have (at least)two solutions (Assume f is differentiable). Furthermore, f00(x) = 0 must have atleast one solution.4. Show that, if f(x) is increasing then 1/f(x) is decreasing.5. Explain the first and second derivative test. What are they testing for?6. State the three “Value Theorems”:7. Compute ∆y and dy for the given x and dx = ∆x. Sketch a diagram and label ∆x, ∆yand dy, if f(x) = 6 − x2, x = −2, ∆x = 1.8. Linearize at x = 0:y =√xex2(x2+ 1)109. Estimate by linear approximation the change in the indicated quantity.(a) The volume, V = s3of a cube, if its side length s is increased from 5 inches to 5.1inches.1(b) The volume, V =43πr3of a sphere, if its radius changes from 2 to 2.1(c) The volume, V =1000p, of a gas, if the pressure p is decreased from 100 to 99.(d) The period of oscillation, T = 2πqL32, of a pendulum, if its length L is increasedfrom 2 to 2.2.10. For the following problems, find where f is increasing or decreasing. If asked, also checkconcavity.(a) f(x) = 3x4− 4x3− 12x2+ 5 (Also check for concave up/down)(b) f(x) =xx+1(c) f(x) = x√x2+ 111. Show that the given function satisfies the hypotheses of the Mean Value Theorem.Find all numbers c in that interval that satisfy the conclusion of that theorem. Forcomparison purposes, given these functions and intervals, what would the IntermediateValue Theorem conclude? Finally, find the global max and global min for each function.(a) f(x) = x3, [−1, 1](b) f(x) =√x − 1, [2, 5](c) f(x) = x +1x, [1, 5]12. Show that f(x) = x2/3does not satisfy the hypotheses of the mean value theorem on[−1, 27], but nevertheless, there is a c for which: f0(c) =f(27) − f(−1)27 − (−1)Find the valueof c.13. At 1:00 PM, a truck driver picked up a fare card at the entrance of a tollway. At 2:15PM, the trucker pulled up to a toll booth 100 miles down the road. After computing thetrucker’s fare, the toll booth operator summoned a highway patrol officer who issued aspeeding ticket to the trucker. (The speed limit on the tollway is 65 MPH).(a) The trucker claimed that she hadn’t been speeding. Is this possible? Explain.(b) The fine for speeding is $35.00 plus $2.00 for each mph by which the speed limit isexceeded. What is the trucker’s minimum fine?14. Let f(x) =1x(a) What does the Extreme Value Theorem (EVT) say about f on the interval [0.1, 1]?(b) Although f is continuous on [1, ∞), it has no minimum value on this interval. Whydoesn’t this contradict the EVT?15. Let f be a function so that f(0) = 0 and12≤ f0(x) ≤ 1 for all x. Use the Mean ValueTheorem to explain why f(2) cannot be 3.16. Sketch the graph of a function that satisfies all of the given properties:f0(−1) = 0, f0(1) does not exist, f0(x) < 0 if |x| < 1, f0(x) > 0 if |x| > 1f(−1) = 4, f(1) = 0, f00(x) > 0 if x > 0217. Find the local maximums and local minimums of f using both the first and secondderivative tests:f(x) = x +√1 − x18. If f(x) = 3x5−5x3+3, find the intervals of increase or decrease, find the local max/min,find the intervals of concavity.19. Show the equation x4+ 4x + c = 0 has at most 2 real solutions.20. If 3 ≤ f0(x) ≤ 5 for all x, s how that the change f(8) − f(2) is at least 18 and at most30.21. Related Rates Extra Practice:(a) The top of a 25-foot ladder, leaning against a vertical wall, is slipping down thewall at a rate of 1 foot per second. How fast is the bottom of the ladder slippingalong the ground when the bottom of the ladder is 7 feet away from the base ofthe wall?(b) A 5-foot girl is walking toward a 20-foot lamppost at a rate of 6 feet per second.How fast is the tip of her shadow (cast by the lamppost) moving?(c) Under the same conditions as above, how fast is the length of the girl’s shadowchanging?(d) A rocket is shot vertically upward with an initial velocity of 400 feet per second.Its height s after t seconds is s = 400t − 16t2. How fast is the distance changingfrom the rocket to an observer on the ground 1800 feet away from the launch site,when the rocket is still rising and is 2400 feet above the ground?(e) A small funnel in the shape of a cone is being emptied of fluid at the rate of 12cubic centimeters per second (the tip of the cone is downward). The height of thecone is 20 cm and the radius of the top is 4 cm. How fast is the fluid level droppingwhen the level s tands 5 cm above the vertex of the cone [The volume of a cone isV =13πr2h].(f) A balloon is being inflated by a pump at the rate of 2 cubic inches per second. Howfast is the diameter changing when the radius is12inch?(g) A particle moves on the hyperbola x2−18y2= 9 in such a way that its y coordinateincreases at a constant rate of 9 units per second. How fast is the x−coordinatechanging when x = 9?(h) An object moves along the graph of y = f(x). At a certain point, the slope of thecurve is12and the x−coordinate is decreasing at 3 units per second. At that point,how fast is the y−coordinate changing?(i) A rectangular trough is 8 feet long, 2


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