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
๐ฅโโ2 ๐ฅ+2
=
2.
lim ๐(๐ฅ) =
๐ฅโ โ1+
lim ๐ (๐ฅ) =
๐ฅโ โ1โ
lim ๐ (๐ฅ) =
๐ฅโ โ1
lim ๐(๐ฅ) =
๐ฅโ 2
3. Sketch a graph with the following characteristics.
a.
b. :
lim f ( x) ๏ฝ 5 and f (1) ๏ฝ 2
x ๏ฎ1
lim f ( x) DNE and f (1) ๏ฝ 3
x ๏ฎ1
4. Evaluate the following limits.
a) lim
๐ฅ 2 +4๐ฅโ12
๐ฅโ 2
๐ฅ 2 โ2๐ฅ
b) lim
11๐ฅ+5
๐ฅโ+โ 5๐ฅโ8
c)
lim
9โ๐ฅ 4
๐ฅ 2 โ๐ฅ
๐ฅโ 5
๐ฅโ 2
g) lim
h) lim
๐ฅ 2 โ6๐ฅ+9
=
i) lim
๐ฅโ0
=
๐ฅ 2 โ9
2๐ฅ 2 โ1
๐ฅโ1 ๐ฅโ1
๐ฅโ+โ 9๐ฅ 3 โ1
e) lim 9 =
f) lim โ๐ฅ + 2 + 3๐ฅ =
๐ฅโ3
๐ฅโโโ ๐ฅ+1
d) lim
=
=
โ๐ฅ+4โ2
๐ฅ
=
๐ฅโ1
๐ฅ 2 โ1
,๐ฅ > 1
5. If ๐(๐ฅ) = {๐ฅ + 1, โ3 โค ๐ฅ โค 1
โ2, ๐ฅ < โ3
a. lim+ ๐(๐ฅ)=
d. lim + ๐(๐ฅ) =
b. limโ ๐(๐ฅ)=
e. lim โ ๐(๐ฅ)=
c. lim ๐(๐ฅ)=
f. lim ๐(๐ฅ)=
๐ฅโ1
๐ฅโโ3
๐ฅโ1
๐ฅโโ3
๐ฅโ1
๐ฅโโ3
6. Find the value of x that would make f(x) continuous at x = 2
๐ (๐ฅ ) = {
๐ฅ 2 โ ๐๐ฅ, ๐ฅ โค 2
๐ฅ + 4๐, ๐ฅ > 2
7. Determine whether the following are continuous at the given x value. Justify.
๐ฅ 2 +5๐ฅโ6
2
a. ๐ฆ = ๐ฅ + 4๐ฅ + 1 at x = 2
c. ๐ (๐ฅ) = {
,๐ฅ โ 1
at x = 1
4, ๐ฅ = 1
๐ฅ 2 โ1
๐ฅ 2 โ9
, ๐ฅ โ โ3
b. ๐ (๐ฅ) = { ๐ฅ+3
at x = -3
2๐ฅ, ๐ฅ = โ3
d.
at x = 1
8. Use the definition of derivative to find fโ(x).
a. f ๏จ x ๏ฉ ๏ฝ 3x ๏ญ 5
b. f ๏จ x ๏ฉ ๏ฝ x2 ๏ญ 7 x ๏ซ 2
9. Find the derivative of the following functions. Simply your answers!
a. f ๏จ x ๏ฉ ๏ฝ 92
f . y ๏ฝ 3 x5 ๏ญ 6 x
b. y ๏ฝ ๏ญ17 x
g. f ๏จ x ๏ฉ ๏ฝ
c. f ๏จ x ๏ฉ ๏ฝ ๏ญ4 x3 ๏ซ 12 x 2 ๏ญ 3
h. y ๏ฝ 4 ๏ซ
d.y ๏ฝ
3 5
๏ญ
x3 x
12
๏ญ 3x
3
x
1
x
2
e.g ๏จ x ๏ฉ ๏ฝ 2 ๏ซ 3x ๏ญ 5 x
2
i. f ๏จ x ๏ฉ ๏ฝ ๏ญ15 x 4 ๏ญ x 3 ๏ซ
4
x7
10. Find the derivative of the following functions.
a. f ๏จ x ๏ฉ ๏ฝ ๏จ15 x 4 ๏ญ 3 x 2 ๏ฉ๏จ 5 x 2 ๏ญ 8 x ๏ซ 9 ๏ฉ
b. y ๏ฝ 5 x 4 ( x 3 ๏ซ 4 x)
c. f ๏จ x ๏ฉ ๏ฝ
5x ๏ญ 3
๏ญ2 x ๏ซ 7
x4
d .g ๏จ x ๏ฉ ๏ฝ 4
3x ๏ญ 2 x
11. Find the second derivative of the following functions.
a. f ๏จ x ๏ฉ ๏ฝ x ๏ญ
9
x5
๏ญ9
b. y ๏ฝ 3 x 3 ๏ซ x 2
12. Write the equation of the line tangent to the curve at the given point.
a. f ๏จ x ๏ฉ ๏ฝ ๏จ 2 x 2 ๏ญ 3x ๏ซ 4 ๏ฉ๏จ 5 x 4 ๏ญ 3๏ฉ at the point (1, 6)
b. a. f ๏จ x ๏ฉ ๏ฝ 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 ๏จ x ๏ฉ ๏ฝ
8x2
x2 ๏ซ 1
b. y ๏ฝ ๏จ x ๏ญ 2 ๏ฉ ๏จ x 2 ๏ญ x ๏ญ 11๏ฉ
c. f ๏จ x ๏ฉ ๏ฝ
x
x ๏ซ4
2
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