AP BC Calculus First Semester Exam Review Guide



2009–2010 AP BC Calculus First Semester Exam Review Guide

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

▪ Be sure you remember the 2nd part is called the Total Change Theorem

Also, in your justifications, if a problem says it is differentiable at x = a, then the function is continuous at x = a. (Differentiability implies continuity, but not vice-versa.)

II. VOCABULARY/KEY CONCEPTS

CHAPTER 1:

1. Distinguish the graphs of parent functions without the use of a calculator. (handout)

2. Define an absolute value function as a piecewise function. (pg. 17)

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 (pgs. 41–42)

CHAPTER 2:

1. Properties of limits (pgs. 61–62 (x approaches c) and pg. 71 (x approaches ±∞))

2. Show a limit that you are unable to calculate using the Sandwich Theorem (pg. 65 example 9)

3. Find horizontal and vertical asymptotes

4. Continuity

5. Difference between average rates of change and instantaneous rates of change

1. Find tangent lines using the instantaneous rate of change

2. Find normal lines using the instantaneous rate of change

CHAPTER 3:

1. Know the definition of a derivative

▪ 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.

2. Rules for differentiation (pgs. 116–119)

▪ Product rule, Quotient rule, Chain Rule, Chain Rule Parametrically

3. **Displacement, average velocity, instantaneous velocity, speed, acceleration**

4. Derivatives of all trig functions and inverse trig functions (Appendix handout)

5. **Implicit differentiation**

6. Derivatives of e and ln

CHAPTER 4:

1. **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,

2. **Graphing using f’ and f’’**

3. **Optimization**

▪ Maximize area/volume

▪ Maximize Profit

▪ Minimize Cost

4. **Related rates**

CHAPTER 5:

1. **Use Riemann sums (RRAM, LRAM, and MRAM) without a calculator**

2. Properties of definite integrals (pg. 285)

3. Average Value (pg. 287)

4. **Use the FTC**

5. Know the difference between

▪ Net area ( area below the x-axis is counted as negative

▪ Total area ( area below the x-axis is counted as positive (the general word area means to find total area)

6. 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))

CHAPTER 6:

1. **Draw a slope field**

2. **Use Euler’s method to approximate a solution without knowing the equation**

3. Use substitution to evaluate an indefinite integral

4. **Antiderivatives** (from Appendix sheet):

▪ 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

5. **Integration by parts**

6. Tabular integration

7. Solving separable differential equations

8. Law of exponential change

▪ Compounding Interest annually, monthly, weekly, etc.

▪ Continuously Compounding Interest

9. Half Life

10. **Integration by partial fractions**

11. **Logistic Growth**

III. SUGGESTED REVIEW PROBLEMS (from the book)

CHAPTER 1:

▪ Pg. 35 (#17, 19, 21)

CHAPTER 2:

▪ Pg. 66 (#23, 25, 38, 50,)

▪ Pg. 76 (#11,)

▪ Pgs. 84 (#19)

▪ Pgs. 92-93 (#2, 5, 31)

CHAPTER 3:

▪ Pg. 106 (#19)

▪ Pg. 124 (#11, 19, 23, 37, 41)

▪ Pgs. 136-138 (#15, 19, 27)

▪ Pg. 146 (#23, 25)

▪ Pgs. 153-154 (#17, 19, 25, 43)

▪ Pgs. 162-163 (#11, 17, 23)

▪ Pg. 170 (#23, 25, 27)

▪ Pgs. 178-179 (#7, 19, 53)

CHAPTER 4:

▪ Pg. 194 (#13, 17)

▪ Pg. 202 (#2, 11)

▪ Pgs. 215-216 (#7, 31, 35, 37)

▪ Pgs. 226-229 (#9, 11, 23, 25)

▪ Pgs. 251-254 (#11, 15, 17, 19, 27)

CHAPTER 5:

▪ Pgs. 270-273 (#15, 19, 33)

▪ Pgs. 291-292 (#5, 31, 3547, 49)

▪ Pgs. 302-304 (#7, 11, 17, 29, 31, 35, 37, 47, 51, 59)

▪ Pgs. 312-314 (#5, 11, 35)

CHAPTER 6:

▪ Pgs. 373-375 (#3, 7, 11, 15, 19, 23, 27, 37, 43, 51, 55, 59, 63, 65)

IV. EXTRA REVIEW PROBLEMS:

1. The graph of [pic] is given [pic]

Find the slope of this curve at the point [pic]

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 [pic], 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 [pic]

4. [pic] 5. [pic]

6. [pic] [pic]

7. [pic] 8. [pic]

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?

Use partial fractions to evaluate the integrals:

10. [pic]

Use integration by parts to evaluate:

11. [pic] 12. [pic] 13. [pic]

Evaluate the integrals:

14. [pic] 15. [pic] 16. [pic]

17. [pic] 18. [pic]

19. Graph the equation given parametrically by the equations [pic] and [pic]. 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?

25. The ACME company has begun selling an MP3 player called the FooPod at a price of $200 – 0.05x, where x is the number of FooPods ACME produces each day. The p

arts and labor cost for each FooPod is $140, and marketing and operational costs amount to an additional $9500 per day. Approximately how many FooPods should ACME produce and sell each day to maximize profit?

26. Assume f(x) is a continuous function and the chart below represents a selection of its function values. Estimate the average value of f(x) on the interval [–3, 7] using the trapezoidal rule.

|x |–3 |–1 |1 |3 |5 |7 |

|y |6 |8 |9 |4 |–1 |–5 |

27. Seven hours after a community water tank is filled, monitoring equipment reports that water is leaking from the tank at a rate of [pic] gallons per hour (where t is the number of hours elapsed since the tank was last filled). Calculate the total amount of water that leaked out of the tank during those seven hours.

28. At the start of a scientific experiment, a scientist observes 125 bacterial colonies growing in the agar of a Petri dish. Exactly six hours later, the number of colonies has grown exponentially to 190.

a.) Approximately how many bacterial colonies will be present exactly one day after the experiment begins?

b.) Assuming that resources are not a limiting issue to growth, how many hours will it take the total bacteria population to reach 20,000?

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