AP CALC MIDTERM REVIEW 2015
Chapter 1 Questions
1. Evaluate the limit: lim
$→&
$ ' ($)*
&)$
2. Evaluate the limit, if it exists: lim
$→+
3. Evaluate the limit, if it exists: lim
$),)&
$)+
- )
. '
$→& $)&
4. Evaluate the limit, if it exists: lim
0123- $
$→/ 4523- $
5. Estimate the limit, if it exists: lim 𝑓(𝑥) where f(x) is represented $→6
by the given graph: sin 2𝑥, 𝑥 ≤ 𝜋
what value of k will 2𝑥 + 𝑘, 𝑥 > 𝜋
make this piecewise function continuous? 6. Given the function: 𝑓 𝑥 =
/
7. Find the limit: lim 𝑥(𝑒 $ + ) $→F
$
8. Identify the vertical asymptotes for 𝑓 𝑥 =
$ ' (6$)H
$ ' ($)&
AP CALC MIDTERM REVIEW 2015
9. If 𝑝(𝑥) is a continuous function on a closed interval [1,3], with 𝑝 1 ≤ 𝐾 ≤ 𝑝(3), and c is in the closed interval [1,3], then which of the following statements must be true? a. 𝑝 𝑐 =
N 6 (N(/)
&
N 6 )N(/)
b. 𝑝 𝑐 =
&
c. There is at least one value c, such that p(c)=K d. There is only one value c, such that p(c)=K e. c=2 10. How many vertical asymptotes exist for the function 𝑓 𝑥 =
/
in the open interval 0 < 𝑥 < 2𝜋 ? & 452' $)452 $)/
Chapter 2 Questions
Q2
1. What does the limit statement lim
a. 0 $→/
$(/ )Q2 &
$)/
represent? R
b. ln 𝑥 + 1 R$
c. 𝑓 S 1 , 𝑖𝑓 𝑓 𝑥 = ln 𝑥 + 1 d. 1 e. The limit does not exist 2. Find the derivative of the function: 𝑦 =
H
$V
RW
3. Find if 3𝑥𝑦 = 4𝑥 + 𝑦 & R$
4. If the nth derivative of y is denoted as 𝑦
𝑦 [ is the same as: Y
and 𝑦 = − sin 𝑥, then AP CALC MIDTERM REVIEW 2015
a. 𝑦
RW
b. c. R$
R'W
R$ '
RVW
d. V
R$
e. None of the above
5. Find the second derivative of 𝑓 (𝑥) if 𝑓 𝑥 = 2𝑥 + 3 H RW
6. Find for 𝑦 = 4 sin& 3𝑥 R$
7. In right triangle ∆𝐴𝐵𝐶, point A is moving along a leg of the right triangle toward point C at a rate of .5 cm/sec and point B is moving toward point C at a rate of 1/3 cm/sec. What is the rate of change of the area of ∆𝐴𝐵𝐶, with respect to time, at the instant when AC = 15 cm and BC = 20cm? 6
8. If 𝑓 2 = −3, 𝑓 S 2 = , 𝑎𝑛𝑑 𝑔 𝑥 = 𝑓 )/ (𝑥), what is the H
equation of the tangent line to 𝑔 𝑥 𝑤ℎ𝑒𝑛 𝑥 = 3 ? Chapter 3 Questions
1. Let M represent the absolute maximum of f(x) in an interval. Let R represent a root of f(x) in the given interval. Let m represent the absolute minimum of f(x) in the interval. If 𝑓 𝑥 = 𝑥 6 − 3𝑥 & , then which of the following is true over the closed interval −3 ≤ 𝑥 ≤ 1? a. M and R occur at a critical point and m occurs at an endpoint. b. M and m occur at critical points. AP CALC MIDTERM REVIEW 2015
c. M, m, and R occur at endpoints of the given interval. d. M occurs at an endpoint, whereas m and R occur at a critical point. e. M and R occur at an endpoint, where as m occurs at a critical point. 2. What value of c in the open interval (0,4) satisfies the Mean Value Theorem for 𝑓 𝑥 = 3𝑥 + 4 ? S
3. If 𝑓 𝑥 =
$(/ $ '
$)/
V
, then on which interval(s) is the continuous function f(x) increasing? 4. The points of inflection for f(x) are at x=p1 and x=p2. Which of the following is(are) true? a. The points of inflection for f(x-­‐a) are at x=p1+a and x=p2+a b. The points of inflection for bf(x) are at x=bp1 and x=bp2 c. The points of inflection for f(cx) are at x=p1/c and x=p2/c 5. Evaluate lim
$ ' )/H
$→f 6)&$
6. The graph of f’(x) is given below. On which interval(s) is the function f(x) both increasing and concave up? AP CALC MIDTERM REVIEW 2015
7. A farmer has 100 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens, as shown in the diagram. All three pens have the same width. Which value of y produces a maximum total fenced area? 8. For the function 𝑓 𝑥 = 12𝑥 , − 5𝑥 H , how many of the inflection points are also extrema? a. 4 b. 3 c. 2 d. 1 e. none 9. The position of an object moving along a straight line for 𝑡 ≥ 0 is given by 𝑠/ 𝑡 = 𝑡 6 + 2, and the position of a second object moving along the same line is given by 𝑠& 𝑡 = 𝑡 & . If both objects begin at t=0, at what time is the distance between the objects a minimum? 10. Given the following conditions for f(x), which graph best illustrates f(x)? The domain of f(x) are the real numbers except x=1. lim 𝑓 𝑥 = −1; lim3 𝑓 𝑥 = ∞; limm 𝑓 𝑥 = −∞ . $→)f
$→/
𝑓 S 𝑥 > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 1, $→/
AP CALC MIDTERM REVIEW 2015
𝑎𝑛𝑑 𝑓 S 𝑥 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑎𝑡 𝑥 = 1. 𝑓"(x)>0 for x<1, f" 𝑥 < 0 𝑓𝑜𝑟 𝑥 > 1, 𝑎𝑛𝑑 𝑓"(𝑥) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑎𝑡 𝑥 = 1. AP CALC MIDTERM REVIEW 2015
Chapter 4 Questions
1. sec & 𝑥 − 2 𝑑𝑥 2. −
H
y
$
$z
V −
𝑑𝑥 3. 4. 5. Approximate the area under the curve over the given interval using 5 right endpoint rectangles. AP CALC MIDTERM REVIEW 2015
6. Find the average value of the function over the given interval. Then, find the values of c that satisfy the Mean Value Theorem for Integrals. AP CALC MIDTERM REVIEW 2015
Chapter 1 Answers 1. -­‐5 2. ¼ 3. -­‐1/4 {
4. &{(H
5. The limit does not exist 6. −2𝜋 7. 1 8. x=-­‐2 9. C 10. 3 Chapter 2 Answers 1. C /&
2. − | 3. $
6W)H
&W)6$
4. D 5. 48 2𝑥 + 3 & 6. 24 sin 3𝑥 cos 3𝑥 7. −7.5 𝑐 𝑚& /𝑠𝑒𝑐 Chapter 3 Answers 1. A 2. 5/3 3. (−∞, −1) ∪ (1, ∞) 4. a and c 5. -­‐1/2 6. (−2, −1) ∪ (0,1) AP CALC MIDTERM REVIEW 2015
7. 25/3 8. e 9. 2/3 10. e Chapter 4 Answers 1. tan 𝑥 − 2𝑥 + 𝐶 &
&
2. ' + | + 𝐶 $
$
3. 12 4. .414 5. 25 6. Average Value -­‐2, Values of C -­‐5, -­‐3