CALCULUS I: REVIEW SHEET FOR THE FINAL EXAM PART I: Expect one or two questions from the following topics: Problem #1. Give the precise definition of the derivative of a function. Then compute the derivative of the following functions using the definition of the derivative. (1) f (x) = x + e π
x (2) f (x) = 2x+1
2
Problem #2. Use linear approximations (a.k.a linearizations) to explain why the following hold: (1) Product Rule; (2) Chain Rule; (3) Quotient Rule. Problem #3. State and explain the Fundamental Theorem of Calculus (parts I and II). Then state the Net Change Theorem. PART II: Expect one question from the following topics: Problem #4. Please find the following: √ (1) the approximation of x around x = 100 using the cubic Taylor polynomial. √ Also find an approximate value of 98; √ (2) the approximate value of 3 9 using the cubic Taylor polynomial for the √ function 3 x around x = 8; (3) the linearization of the function f (x) = sin−1 (x) around x = 1 ; 2 (4) the approximate value of tan−1 (.1) using the quadratic Taylor approximation of tan−1 (x) around x = 0; 2 x (5) the cubic Taylor approximation of the function F (x) = 0 e−t dt around x = 0. Problem #5. Use Riemann sums to find an underestimation and an overestimation of the following definite integrals.
1
(1)
0 1
e−x dx; suggested value of the number segments is n = 5. e−x dx; suggested value of the number of segments is n = 4.
−1
2
2
(2)
PART III: Expect one or two questions from the following topics: Problem #6. Please graph the following functions by hand and then check your answer on a graphing calculator. Make sure you address vertical and horizontal asymptotes, the intervals of increase and decrease, maxima & minima and intervals of concavity. (1) f (x) = 3x3 − 449x2 − 200x;
ln x (2) f (x) = 20−ln x ; −999 (3) f (x) = ee −1001 . x
1
x
2
Problem #7. Find the following:
x −4 (1) the vertical asymptotes of f (x) = ln(x−1) ;
2
(2) the equation of the tangent line to the curve y = tan(x) at the point ( π , 1); 4 sin(x) − x x3 (4) the area enclosed by the parabola y = −x2 + 6x − 8 and the x-axis; e t+1 (5) the definite integral dt; t 1 (3) the limit lim
x→0
(6) the equation of the tangent line to the hyperbola 5 ( 2 , 1);
x2 4
− 9y = 1 at the point 16
2
(7) the function f (x) for which satisfies f (x) = cos(πt) and f (0) = f (0) = 0; (8) the antiderivative 1 dx; 1 + (2x + 1)2 1 (9) the horizontal asymptotes of the function f (x) = (1 + x )x ;
(10) the area between the line y = x and the parabola y = x2 .
PART IV: Expect two to three questions from the following topics: Problem #8. A rectangular plot of land will be fenced into three equal portions by two dividing fences parallel to two sides. If the area to be enclosed is 4000 m2 , find the dimensions of the land that require the least amount of fence. Problem #9. A box with an open top is to be constructed from a rectangular piece of cardboard 1 m wide and 2 m long by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Problem #10. A rain gutter with a rectangular cross section is made from a 1-ft × 20-ft piece of metal by bending up equal amounts from the 1-ft side. How should the metal be bent up on each side in order to make the capacity (i.e volume) of the gutter a maximum? Problem #11. A jewelry box is to be made in the shape of a rectangular box with a square base. The volume of the box needs to be 250 cubic inches. The material used to manufacture the bottom side of the box costs $1 per square inch. The material for the four sides costs $2 per square inch and the material for the lid costs $7 per square inch. Find the dimensions that will minimize the cost of the material to manufacture the box. Problem #12. A can is to be made to hold 2π liters of oil. The metal used to manufacture the base costs .8 cents per unit area (in our case dm2 ) and the metal used to manufacture the side costs 2.7 cents per unit area. Find the dimensions that will minimize the cost of the metal to manufacture the can.
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Problem #13. Of all rectangles with perimeter 20 inches, find the one with the shortest diagonal. Problem #14. A particle moves on a straight line. Its (instantaneous) velocity at time t is described by the function π v(t) = 2 sin (2πt + ). 2 Here time is measured in minutes and velocity is measured in inches per minute. The motion started from the position x = 0 at the moment t = 0. (1) Find the position x(t) of the particle at t minutes. (2) What is the net change in position and what is the average velocity of the 1 particle during the time interval from t = 2 to t = 1? (3) During what time intervals is the particle moving in positive direction, and during what time intervals is the particle moving in the negative direction? (4) Find the total distance traveled by the particle during the time interval between t = 0 and t = 1. Problem #15. A car is traveling at 31 m and the driver slams on the breaks, s m causing the car to skid and decelerate at a constant rate of 5.55 s2 until it stops. Find the velocity of the particle as a function of time. How long does it take for the car to stop? How far does the car travel before stopping? Problem #16. The following graph plots acceleration vs. time for a particle moving in one dimension. Graph velocity vs. time for the particle, assuming zero 2.4 velocity at time t = 0. (Hint: You need to graph v(t) from knowing the graph of v (t).)
1.6
0.8
8
-4
-3.2
-2.4
-1.6
-0.8
0
0.8
1.6
2.4
3.2
4
4.8
-0.8
cm Problem #17. The length l of a rectangle is decreasing at the rate of 2 sec while cm -1.6 the width w is increasing at the rate of 2 sec . When l = 12 cm and w = 5 cm, find the rates of change of (1) the area; -2.4 (2) the perimeter; (3) the lengths of the diagonals of the rectangle.
Problem #18. A car is moving west towards an intersection at the speed of 25 mph. A biker is approaching the intersection from the south at the speed of 7 mph. At what rate are the car and the biker approaching each other at the moment when the biker is .1 miles from the intersection and the car is .3 miles from the intersection?
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Problem #19. Air is being pumped into a spherical balloon at the rate of 3 64 inches . At what rate is the radius of the balloon increasing at the moment sec when there is 729 inches3 of air in the balloon?
ft Problem #20. A man 6 ft tall walks at the rate of 5 sec toward a streetlight that is 16 f t above the ground. At what rate is the tip of the shadow moving? At what rate is the length of his shadow changing?
Problem #21. A blown-out undersea oil well is spewing oil at 25 m3 per second, forming a circular slick about 0.01 m thick on the surface of the calm sea. Compute the rate of increase of the radius of the slick at the moment when the radius is r = 10 m. Problem #22. Your favorite branch of your favorite bank is open from 9AM to 6PM. The number of customers the branch had during the first t hours of the work day is given by 27 2 f (t) = t − t3 . 2 You would like to go do some business in the bank, but you don’t want to wait too much. You’d like to avoid the moment during the bank’s hours when the rate at which the people are getting into the bank is biggest. Find that moment. Problem #23. A cat in the window is 16 m away from a busy street. A woman m and her scary dog are walking down the street at the speed of 1 sec . Find out how fast the dog is approaching the cat at the moment when the cat notices the dog - when the dog is 20 m away from the cat. Is the rate at which the dog is approaching the cat getting bigger or smaller as the time progresses? Problem #24. You have a big sheet of wrapping paper in the shape of the region x enclosed by the curve y = e− 10 and the lines x = 0, x = 5, y = 0. (1) What is the area of the big sheet of paper? (2) You’d like to cut the big sheet of paper into two pieces of equal area. Explain how the cut should be made, assuming it will be made along a line of the form x = c.