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
o Definition of continuity
o Limit definition of derivatives
o Simplifying and solving limits (also graphically)
o Infinite limits (DNE)
o Limits as x approaches infinity (horizontal asymptotes)
o Intermediate Value Theorem
 Unit 2 – Derivatives
o Basic rules of derivatives
o Implicit differentiation
o Rate of change (average and instantaneous)
o Inverse trig functions
o Derivatives of log/ln and exponential functions
o Mean Value Theorem
o Equation of tangent line
o Parametric, polar, vector derivatives
 Unit 3 – Applications of Differentiation (Ch. 3)
o L’Hopitals
o Extrema and 1st and 2nd Derivative Tests (critical points)
o Extreme Value Theorem
o Curve Sketching (increasing/decreasing, concavity)
o Relationship between graphs of f, f ‘, and f “
o Tangent Line Approximation
o Optimization
o Related Rates
o Rectilinear motion – position, velocity, acceleration
 Unit 4 – Integration (Ch. 4)
o Antiderivatives
o Properties
o Fundamental Theorem of Calculus (two parts)
o U-substitution
o Average value (versus average rate of change from Unit 2)
o Given graph of f ‘, characteristics of f and f “ graph
o Numerical integration – Riemann sums, Trapezoidal sums, Simpson’s rule
o Parametric, polar, vector integration
o Position, velocity, acceleration integration
 Unit 5 – Techniques of Advanced Integration (Ch. 8)
o Different methods (completing the square, add/subt. same number, etc.)
o By parts (tabular method; LIPET)
o Trig identities
o Trig substitution
o Partial fractions
o L’Hopitals
o Improper Integrals
Format of Midterm (AP style)


Section I
o Part A – 20 minutes, 10 multiple choice questions, no calculator
o Part B – 30 minutes, 10 multiple choice questions, calculator okay
Section II
o Part A – 30 minutes, 2 free response, calculator okay
o 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 f ' ( x)  ln ( x  2) , then the graph of y  f (x) is decreasing if and only if
a) 2  x  3
b) 0  x
c) 0  x  1 d) x  1
e) x  2
_____ 2) For x  0 , the slope of the tangent to y  x cos ( x) equals zero whenever
a) tan ( x)   x
b) tan( x)  1x
c) tan ( x )  x
d) sin ( x)  x
e) cos ( x)  x
_____ 3) The function F is defined by F ( x)  G x  G( x) where the graph of the
function
G is shown below. Find the approximate
value of F ' (1) .
a) 73
b) 23
c)  2
d)  1
e)  23
6
_____ 4)
1
  x
2

 2 x  dx 

a) ln 4  32
c) ln 3  32
e) ln12  32
b) ln 3  40
d) ln 4  40
_____ 5) Find an equation of the line tangent to the graph of y  x 3  3x 2  2 at its
point of inflection.
a) y   3x  1
b) y   3x  7
c) y  x  5
d) y  3x  1
e) y  3x  7
_____ 6)
 cos 3 
2 x dx 
b)  sin 3  2 x   C
a) sin 3  2 x   C
c)
1
2
sin 3  2 x   C
e) 
_____ 7) What is
1
5
d) 
1
2
sin3  2x  C
sin 3  2x  C
9x 2  2
lim
x  
4x  3
a)
3
2
b)
c)
2
3
d) 1
?
3
4
e) The limit does not exist.
x2
_____ 8) Suppose F ( x) 
2
0
1
dt for all real x. Find F ’(-1).
 t3
a) 2
b) 1
d)
e) 
1
3
c) -2
2
3
_____ 9) What is the average value of 2t 3  3t 2  4 over the interval  1  t  1 ?
a) 0
b)
7
4
c) 3
d) 4
e) 6
_____ 10) Find
lim
x  1
x  1
.
x  1
a) 0
b)
1
2
c) 1
d)
3
2
e) The limit does not exist.
_____ 11) If y 
a)
b)
c)
d)
e)
cos2 ( x)  sin2 ( x) , find y' .
1
0
 2cos ( x)  sin ( x)
2cos ( x)  sin ( x)
 4 cos ( x) sin ( x)
_____ 12) Find the area under the graph of y  4 x 3  6 x 
a) 32  ln 2 units2
1
x
on the interval 1  x  2 .
b) 30  ln 2 units2
c) 24  ln 2 units2
d) 994 units2
e) 21 units2

_____ 13) Evaluate
x  2
dx .
x 1
a)  x ln x  1  C
b) x  ln x  1  C
c) x  ln x  1  C
d) x 
x 1  C
e) x 
x 1  C
_____ 14) Find an equation for a tangent to the graph of y  Arc tan
(There is one like this on the midterm!)
a) x  3 y  0
b) x  y  0
c) x  0
d) y  0
e) 3x  y  0
_____ 15) Find


d
ln e 3 x .
dx
a) 1
b) 3
d)
e)
1
e3 x
3
e3 x
c) 3x
x
at the origin.
3
_____ 16) The acceleration at time t  0 of a particle moving along the x-axis is
a(t )  3t  2 ft sec 2 . If at t  1 seconds the velocity is 4 ft sec and the position is x  6 ft, find
the position x(t ) at t  2 seconds.
a) 8 ft
b) 11 ft
c) 12 ft
d) 13 ft
e) 15 ft
_____ 25) Find the approximate value of y 
to the graph at x  0 .
a) 2.01
b) 2.02
c) 2.03
d) 2.04
3  e x at x  0.08 , obtained from the tangent
e) 2.05
_____ 26) The function f is defined on the interval  5,5 and
its graph is shown to the right. Which of the
following statements are true?
I.
lim
x  1
f ( x)   1
II.
lim
h  0
f (2  h)  f (2)
 2
h
lim
f ( x)  f (3)
III.
x  1
a) I only
c) I and II only
e) I, II, and III
b) II only
d) II and III only
More practice
1  cos 2 x
x 0
x
1.
Find lim
a. 1
b. 0
c. 
d. Does not exist
e. None of these
b.0
c.  
d. Does not exist
e. None of these
1
x 0 x
2.
Find lim
a. 
a  bx 4
x cx 4  x 2
3.
Find the limit: lim
b. 
a. 0
4.
b
c
d.
a
c
e. None of these
Determine the sign of the second derivative at the indicated point:
a. Zero
d. Negative
b. Undefined
e. None of these
c. Positive
Find y if y2  3xy  x2  7
5.
a.
6.
2x  y
3x  2 y
Find
a.
7.
c. 
b.
3y  2x
2 y  3x
dy
for y  ln x 2  4
dx
x
2x
b.
x2  4
x2  4
Find the derivative of y = arctan
a.
sec2 x
2 x
b.
1
2 x (1  x)
c.
2x
3 2y
d.
2x
y
e. None of these
c.
x
x 4
d.
1
x
e. None of these
2
x
c.
1
2 x (1  x)
d.
1
(1  x)
e. None of these
8.
Evaluate
 xe dx
x
a. xe x  e x  C
d. xe x  2e x  C
9.
b. xe x  e x  C
e. None of these
c. xe x  2e x  C
Which integral represents the trig substitution form of the integral
a.
 36sin d
 6 cosd
b.
2
d.
10. Evaluate lim
x 0
11. Evaluate
c.
2
36  x 2
dx
 tand
e. None of these
sin( 2 x )
sin( 3 x )
b. 
a. 0
 36cos d

x2
2x  3
 9 x
2
c.
2
3
d.
3
2
e. None of these
dx
1
x
arctan  C
3
3
e. None of these
a. ln(9  x 2 )  C
c. ln(9  x 2 )  arctan
b.
d. ln 3  x  C
x
C
3
12. Find the arc length x  t 2 , y  2t 2  1 , on interval [1, 4]
a. 30 5
b. 75
c. 15 5
d. 5 5
e. None of these
13. Find the slope of the tangent for the curve r  2cos3 at the point where  
a.  6
b.
1
3
c.  3
14. Calculate the area inside the cardioid 1  cos  .
3
3
a. 3
b.
c.
4
2

6
.
d. 1
e. None of these

2
e. None of these
d.
15. Calculate the distance around the graph of the polar curve r  3sin  .
3
a. 6
b.
c. 16
d. 3
e. None of these
2