AP Calculus BC Midterm Review Problems

Topics covered: (Bold are the “newer” material not from Math Analysis or Calc AB)

  • Unit 1 – Limits and continuity
  • Definition of continuity
  • Limit definition of derivatives
  • Simplifying and solving limits (also graphically)
  • Infinite limits (DNE)
  • Limits as x approaches infinity (horizontal asymptotes)
  • Intermediate Value Theorem
  • Unit 2 – Derivatives
  • Basic rules of derivatives
  • Implicit differentiation
  • Rate of change (average and instantaneous)
  • Inverse trig functions
  • Derivatives of log/ln and exponential functions
  • Mean Value Theorem
  • Equation of tangent line
  • Parametric, polar, vector derivatives
  • Unit 3 – Applications of Differentiation (Ch. 3)
  • L’Hopitals
  • Extrema and 1st and 2nd Derivative Tests (critical points)
  • Extreme Value Theorem
  • Curve Sketching (increasing/decreasing, concavity)
  • Relationship between graphs of f, f ‘, and f “
  • Tangent Line Approximation
  • Optimization
  • Related Rates
  • Rectilinear motion – position, velocity, acceleration
  • Unit 4 – Integration (Ch. 4)
  • Antiderivatives
  • Properties
  • Fundamental Theorem of Calculus (two parts)
  • U-substitution
  • Average value (versus average rate of change from Unit 2)
  • Given graph of f ‘, characteristics of f and f “ graph
  • Numerical integration – Riemann sums, Trapezoidal sums, Simpson’s rule
  • Parametric, polar, vector integration
  • Position, velocity, acceleration integration
  • Unit 5 – Techniques of Advanced Integration (Ch. 8)
  • Different methods (completing the square, add/subt. same number, etc.)
  • By parts (tabular method; LIPET)
  • Trig identities
  • Trig substitution
  • Partial fractions
  • L’Hopitals
  • Improper Integrals

Format of Midterm (AP style)

  • Section I
  • Part A – 20 minutes, 10 multiple choice questions, no calculator
  • Part B – 30 minutes, 10 multiple choice questions, calculator okay
  • Section II
  • Part A – 30 minutes, 2 free response, calculator okay
  • Part B – 30 minutes, 2 free response, no calculator

Free Response Practice Problems: Use previous reviews and tests and all the previous AP practice problems you’ve been given.

Multiple Choice Practice Problems (Disclaimer: These are just some practice problems. They do not cover everything on the midterm. You should try these but also study more. Use your unit tests and reviews for those unit tests as a study guide. Also, some of these problems might be repeats from previous tests/reviews that you’ve been given…sorry)

NO CALC

CALCULATOR OKAY

NO CALC

_____ 1) If , then the graph of is decreasing if and only if

a) b)

c) d)

e)

_____ 2) For , the slope of the tangent to equals zero whenever

a)

b)

c)

d)

e)

_____ 3) The function F is defined by where the graph of the function

G is shown below. Find the approximate

value of .

a)

b)

c)

d)

e)

_____ 4)

a) b)

c) d)

e)

_____ 5) Find an equation of the line tangent to the graph of at its point of inflection.

a)

b)

c)

d)

e)

_____ 6)

a) b)

c) d)

e)

_____ 7) What is ?

a) b)

c) d) 1

e) The limit does not exist.

_____ 8) Suppose for all real x. Find F ’(-1).

a) 2b) 1c) -2

d) e)

_____ 9) What is the average value of over the interval ?

a)0

b)

c)3

d)4

e)6

_____ 10) Find .

a) 0b)

c) 1d)

e) The limit does not exist.

_____ 11) If , find .

a)

b) 0

c)

d)

e)

_____ 12) Find the area under the graph of on the interval .

a)

b)

c)

d)

e)

_____ 13) Evaluate .

a)

b)

c)

d)

e)

_____ 14) Find an equation for a tangent to the graph of at the origin.

(There is one like this on the midterm!)

a) b)

c) d)

e)

_____ 15) Find .

a) 1b) 3c)

d) e)

_____ 16) The acceleration at time of a particle moving along the x-axis is . If at seconds the velocity is and the position is ft, find the position at seconds.

a) 8 ftb) 11 ft

c) 12 ftd) 13 ft

e) 15 ft

_____ 25) Find the approximate value of at , obtained from the tangent

to the graph at .

a) 2.01b) 2.02

c) 2.03d) 2.04

e) 2.05

_____ 26) The function f is defined on the interval and

its graph is shown to the right. Which of the

following statements are true?

a) I onlyb) II only

c) I and II onlyd) II and III only

e) I, II, and III

More practice

1.Find

a. 1b. 0c. d. Does not existe. None of these

2.Find

a. b.0c. d. Does not existe. None of these

3.Find the limit:

a. 0b. c. d. e. None of these

4.Determine the sign of the second derivative at the indicated point:

a. Zerob. Undefinedc. Positive

d. Negativee. None of these

5.Find if

a. b. c. d. e. None of these

6.Find for

a. b. c. d. e. None of these

7.Find the derivative of y = arctan

a. b. c. d. e. None of these

8.Evaluate

a.b. c.

d.e.None of these

9.Which integral represents the trig substitution form of the integral

a.b. c.

d.e.None of these

10.Evaluate

a.0b.c.d.e.None of these

11. Evaluate

a. b. c.

d. e. None of these

12. Find the arc length , on interval [1, 4]

a. 30b. 75c. 15d. 5e. None of these

13. Find the slope of the tangent for the curve at the point where .

a. 6b. c. d. 1e. None of these

14. Calculate the area inside the cardioid .

a. b. c. d. e. None of these

15. Calculate the distance around the graph of the polar curve .

a. b. c. 16d. e. None of these