BC Calculus Exam Review
I. “BIG 7” THEOREMS (be able to state and use theorems especially in justifications)
Intermediate Value Theorem Extreme Value Theorem
Rolle’s Theorem Mean Value Theorem for Derivatives & Definite Integrals
FUNDAMENTAL THEOREM OF CALCULUS (proof)
II. KEY CONCEPTS
1. Distinguish the graphs of parent functions without the use of a calculator.
2. Define an absolute value function as a piecewise function.
3. Parametric Equations
§ Be able to graph a curve (be sure to indicate direction on the curve)
§ Be able to write a Cartesian Equation for a curve
4. Properties of logarithms and natural logarithms and change of base formula
5. Properties of limits
6. Show a limit that you are unable to calculate using the Sandwich Theorem
7. Find horizontal and vertical asymptotes
8. Continuity
9. Difference between average rates of change and instantaneous rates of change
10. Find tangent lines using the instantaneous rate of change
11. Find normal lines using the instantaneous rate of change
12. Know the definition of a derivative
13. You will not be asked to find the derivative using the definition, but you might need to justify an open-ended problem or a proof using the definition.
14. Rules for differentiation
15. Product rule, Quotient rule, Chain Rule
16. Displacement, average velocity, instantaneous velocity, speed, acceleration
17. Derivatives of all trig functions and inverse trig functions (Appendix handout)
18. Implicit differentiation
19. Derivatives of e and ln
20. Derivatives of polar
21. Derivatives of parametrics
22. Use critical points to find absolute and relative extrema and points of inflection
· Justify max or min points by stating whether the graph is increasing or decreasing,
· Graphing using f’ and f’’
23. Optimization
· Maximize area/volume
· Maximize Profit
· Minimize Cost
24. Related rates
25. Use Riemann sums (RRAM, LRAM, and MRAM) without a calculator
26. Properties of definite integrals
27. Average Value
28. Use the FTC
29. Know the difference between
· Net area à area below the x-axis is counted as negative
· Total area à area below the x-axis is counted as positive (the general word area means to find total area)
30. Use the Trapezoidal Rule
· Know that on the AP Exam, using “Trapezoidal Sums” does not necessarily mean using the Trapezoidal Rule…the partitions may not be equal, and you may have to find the area of each trapezoid drawn to find the approximate integral. (Area of a trap. = (1/2)(b1 + b2)(h))
31. Simpson’s Rule
32. Use substitution to evaluate an indefinite integral
33. Antiderivatives
· Trig: cos(x), sin(x), sec2x, –csc2 x, sec(x) tan(x), –csc(x) cot(x)
· Exponential: ex, 1/x
· Inverse trig: sin-1x, tan-1x
· Polynomial
· U-sub
· By Parts
· Tabular
34. Differential Equations
· General Solutions
· Initial Value Problems/Particular Solutions
· Separable DE
· Euler’s Method
II. EXAM REVIEW PROBLEMS
1. The parametrization of a curve is given by x = t and y = 1 – t for 0 £ t £ 1. What are the initial and terminal points, if any? Graph the curve indicating direction. Find a Cartesian equation for the curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? ANS: y = –x + 1 (0, 1) to (1, 0)
2. The parametrization of a curve is given by and for 0 £ t. What are the initial and terminal points, if any? Graph the curve indicating direction. Find a Cartesian equation for the curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? ANS: y = –x + 4; x £ 4
3. The parametrization of a curve is given by and for. What are the initial and terminal points, if any? Graph the curve indicating direction. Find a Cartesian equation for the curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? ANS: y = –2x2 + 1; –1 £ x £ 1
4. What is ? ANS: 12 5. Evaluate . ANS: –1
6. What is the average rate of change of over [0, 2]? What is the average rate of change of f(x) over [10, 12]? ANS: over [0, 2] = 1 over [10, 12] = 0.298
7. Find the average rate of change of over . Find the average rate of change over .
ANS: over [π/4, 3π/4] = –1.273 over [π/6, π/2] = –1.654
8. Find the lines that are (a) tangent and (b) normal to the curve at the point (1, 1).
ANS: a.) y = 3x – 2 b.) y = (–1/3)x + (4/3)
9. Suppose u and v are functions of x that are differentiable at x = 0, and that u(0) = 5, u’(0) = –3, v(0) = –1, and v’(0) = 2. Find the values of the following derivatives at x = 0.
a.) b.) c.) d.)
ANS: 13 ANS: –7 ANS: 7/25 ANS: 20
10. On Earth, in the absence of air, a rock thrown vertically upward at a velocity of 24 m/sec reaches a height of meters in t seconds. How high would the rock go? ANS: about 29.388 meters
11. A particle moves along a line so that its position at any time t ≥ 0 is given by the function , where s is measured in meters and t is measured in seconds.
a.) Find the displacement during the first 5 seconds. ANS: 10 m
b.) Find the average velocity during the first 5 seconds. ANS: 2 m/sec
c.) Find the instantaneous velocity when t = 4. ANS: 5 m/sec
d.) Find the acceleration of the particle when t = 4. ANS: 2 m/sec2
e.) At what values of t does the particle change direction? ANS: at t = 3/2 sec
f.) Where is the particle when s is a minimum? ANS: at s = –1/4 m
12. Suppose that the dollar cost of producing x washing machines is .
a.) Find the average cost of producing 100 washing machines. ANS: $110 per machine
b.) Find the marginal cost when 100 machines are produced. ANS: $80 per machine
13. Find the equation of the line tangent to the curve parametrized by and at the point when .
ANS: y = (–1/2)x – (1/2)
14. Find the extreme values of the function over the interval .
ANS: Max value of 3(2/5) at x = –3; min value of 0 at x = 0
15. State whether or not the Mean Value Theorem can be applied for on the interval [0, 1]. If so, find the value of c that satisfies the Mean Value Theorem. ANS: Yes, MVT can be applied. C = 8/27
16. A trucker handed in a ticket at a toll booth showing that in 2 hours, she had covered 159 miles on toll road with speed limit 65 mph. The trucker was cited for speeding. Why? ANS: Because the trucker’s average speed was 79.5 mph and by the MVT, the trucker must have been going that speed at least once during the trip.
17. Your iron works has contracted to design and build a 500-ft3 square based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible. What dimensions should you use?
ANS: 10 ft by 10 ft by 5 ft
18. A spherical balloon is inflated with helium at the rate of 100π ft3/min. How fast is the balloon’s radius increasing at the instant the radius is 5 feet? How fast is the surface area increasing at that instant?
ANS: radius increasing 1 ft/min. SA increasing 40π ft2/min
19. Water is flowing at the rate of 50 m3/min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep? How fast is the radius of the water’s surface changing at that moment? Give your answers in cm/min. ANS: water level falling 1.13 cm/min. Radius changing –8.49 cm/min.
20. The following table gives dye concentrations for a dye-concentration cardiac-output determination. The amount of dye injected in this patient was 5 mg. Use rectangles to estimate the area under the dye concentration curve.
Seconds after injection (t) / 2 / 4 / 6 / 8 / 10 / 12 / 14 / 16 / 18 / 20 / 22 / 24Dye Concentration (adjusted for recirculation (c) / 0 / 0.6 / 1.4 / 2.7 / 3.7 / 4.1 / 3.8 / 2.9 / 1.7 / 1.0 / 0.5 / 0
ANS: 44.8
21. The table below gives data for the velocity of a vintage sports car accelerating from 0 to 142 mi/h in 36 seconds (10 thousandths of an hour).
Time (h) / 0.0 / 0.001 / 0.002 / 0.003 / 0.004 / 0.005 / 0.006 / 0.007 / 0.008 / 0.009 / 0.010Velocity (mph) / 0 / 40 / 62 / 82 / 96 / 108 / 116 / 125 / 132 / 137 / 142
a.) Use rectangles to estimate how far the car traveled during the 36 seconds it took to reach 142 mph.
b.) Roughly how many seconds did it take the car to reach the halfway point? About how fast was the
car going then?
ANS: a.) 0.969 miles b.) 0.006 hour = 21.6 sec. Car going 116 mph
22. Find the average value of the function on [0, π]. ANS: 2/π
23. Find the average value of the function on [–1, 2]. ANS: 4
24. Find of . ANS: (1+x)/(1+x2) 25. Find of . ANS:
26. Find the area of the semielliptical region between the x-axis and the graph of . ANS: 8.886
27. Evaluate . 28. Evaluate . 29. Evaluate .
ANS: 8 ANS: e – 1 ANS: –ln|2 – sin(x)| + C
30. Evaluate 31. Evaluate .
ANS: –ln|cos(ln y)| + C ANS:
32. If and y(–3) = 2, find the particular solution.
ANS: y = ln(t + 4) + 2
III. Ole Miss Exam Review Problems
1. The graph of is given
Find the slope of this curve at the point
2. A hot air balloon rising straight up from a level field is tracked by a range finder 300 ft from the liftoff point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.15 rad/min. How fast is the balloon rising at that moment?
3. Sketch the graph of
4. 5.
6.
7. 8.
9. An open box is to be made by cutting a square from each corner of a 6-inch by 6-inch piece of metal and then folding up the sides. What size square should be cut from each corner to produce a box of maximum volume?
14. 15. 16.
17. 18.
19. Graph the equation given parametrically by the equations and . What is the Cartesian equation for this curve?
20. Estimate the area of the region bounded by the x-axis and the function g(x) = x2 on the interval [0, 5] using a right, left, and midpoint Riemann sum with 4 rectangles of equal width.
21. A policeman clocks a commuter’s speed at 50 mph as he enters a tunnel whose length s exactly 0.75 miles. A second officer measures the commuter’s speed at 45 mph as he exits the tunnel 43 seconds later and tickets the driver for exceeding the posted speed limit of 50 mph. Although the driver was neither exceeding the posted speed limit while entering nor while exiting the tunnel, justify the speeding charge levied by the officer.
22. A baseball is thrown from n initial height of 5 feet above the ground with an initial velocity of 100 ft/sec. Ignoring the effect of wind resistance, the path can be modeled by the equation s(t) = –16t2 + 100t + 5.
a.) What is the velocity of the baseball at t = 1?
b.) What is the acceleration of the baseball at t = 1?
c.) When will the baseball hit the ground?
d.) When does the baseball reach its maximum height?
e.) Assuming the baseball is thrown straight up, what is the maximum height it will reach?
23. Two hikers begin at the same location and travel in perpendicular directions. Hiker A travels due north at a rate of 5 mph. Hiker B travels due west at a rate of 8 mph. At what rate is the distance between the hikers changing 3 hours into the hike?
24. Water stored in an inverted right circular cone leaks out at a constant rate of 2 gallons per day. Assuming the tank is 100 feet high and the radius of its base is 25 feet, at what rate is the depth of the water inside the tank decreasing at the moment it is 40 feet deep?