Name _______________________________________________ PreCalculus Quarterly #4 Review
Be sure to look over previous tests, quizzes, and notes in addition to this review sheet.
1. Evaluate the following limit. Be sure to show all work. lim
4
𝑥→−2 𝑥+2
=
2.
lim 𝑓(𝑥) =
𝑥→ −1+
lim 𝑓 (𝑥) =
𝑥→ −1−
lim 𝑓 (𝑥) =
𝑥→ −1
lim 𝑓(𝑥) =
𝑥→ 2
3. Sketch a graph with the following characteristics.
a.
b. :
lim f ( x)  5 and f (1)  2
x 1
lim f ( x) DNE and f (1)  3
x 1
4. Evaluate the following limits.
a) lim
𝑥 2 +4𝑥−12
𝑥→ 2
𝑥 2 −2𝑥
b) lim
11𝑥+5
𝑥→+∞ 5𝑥−8
c)
lim
9−𝑥 4
𝑥 2 −𝑥
𝑥→ 5
𝑥→ 2
g) lim
h) lim
𝑥 2 −6𝑥+9
=
i) lim
𝑥→0
=
𝑥 2 −9
2𝑥 2 −1
𝑥→1 𝑥−1
𝑥→+∞ 9𝑥 3 −1
e) lim 9 =
f) lim √𝑥 + 2 + 3𝑥 =
𝑥→3
𝑥→−∞ 𝑥+1
d) lim
=
=
√𝑥+4−2
𝑥
=
𝑥−1
𝑥 2 −1
,𝑥 > 1
5. If 𝑔(𝑥) = {𝑥 + 1, −3 ≤ 𝑥 ≤ 1
−2, 𝑥 < −3
a. lim+ 𝑔(𝑥)=
d. lim + 𝑔(𝑥) =
b. lim− 𝑔(𝑥)=
e. lim − 𝑔(𝑥)=
c. lim 𝑔(𝑥)=
f. lim 𝑔(𝑥)=
𝑥→1
𝑥→−3
𝑥→1
𝑥→−3
𝑥→1
𝑥→−3
6. Find the value of x that would make f(x) continuous at x = 2
𝑓 (𝑥 ) = {
𝑥 2 − 𝑘𝑥, 𝑥 ≤ 2
𝑥 + 4𝑘, 𝑥 > 2
7. Determine whether the following are continuous at the given x value. Justify.
𝑥 2 +5𝑥−6
2
a. 𝑦 = 𝑥 + 4𝑥 + 1 at x = 2
c. 𝑓 (𝑥) = {
,𝑥 ≠ 1
at x = 1
4, 𝑥 = 1
𝑥 2 −1
𝑥 2 −9
, 𝑥 ≠ −3
b. 𝑓 (𝑥) = { 𝑥+3
at x = -3
2𝑥, 𝑥 = −3
d.
at x = 1
8. Use the definition of derivative to find f’(x).
a. f  x   3x  5
b. f  x   x2  7 x  2
9. Find the derivative of the following functions. Simply your answers!
a. f  x   92
f . y  3 x5  6 x
b. y  17 x
g. f  x  
c. f  x   4 x3  12 x 2  3
h. y  4 
d.y 
3 5

x3 x
12
 3x
3
x
1
x
2
e.g  x   2  3x  5 x
2
i. f  x   15 x 4  x 3 
4
x7
10. Find the derivative of the following functions.
a. f  x   15 x 4  3 x 2  5 x 2  8 x  9 
b. y  5 x 4 ( x 3  4 x)
c. f  x  
5x  3
2 x  7
x4
d .g  x   4
3x  2 x
11. Find the second derivative of the following functions.
a. f  x   x 
9
x5
9
b. y  3 x 3  x 2
12. Write the equation of the line tangent to the curve at the given point.
a. f  x    2 x 2  3x  4  5 x 4  3 at the point (1, 6)
b. a. f  x   5 3 x at x = 8
13. Find the point(s) at which the graph of the function has a horizontal tangent line (slope = 0).
a. f  x  
8x2
x2  1
b. y   x  2   x 2  x  11
c. f  x  
x
x 4
2