MATH 1910
TEST 1 Review
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1. Evaluate the limits in parts a through m analytically.
x+ x
x 2 − 5x − 6
x −5 x
a. lim
b. lim
c. lim
2
x →4 3x + 1
x →6
x →25 3x − 75
x − 36
5x + 1 x > −2
x+2
e. lim+ f (x), f (x) =  2
f. lim −
x →−2
x →−1 x + 1
x ≤ −2
 x
x −1
sin (11x )
x+6
€
h. lim
i. lim
j. lim+
x→1 x − 1
x →0
x →−6 x + 6
13x
6x − 3 x > 3
x 2 − 2x + 1
l. lim€ f (x), f (x) =  2
m. lim
x→3
x →−1
x + 2 x ≤ 3
x +1
4 + sin x
x →0 cos(x + π )
[ x] + 2
g. lim
x→1 x + 1
[ x]
k. lim−
x →3 x − 1
d. lim
2. Evaluate the limits in parts a and b graphically.
a. lim f (x)
b. lim f (x)
x →2
x →2
y
y
x
x
3. Determine the values of x for which the function is discontinuous and label each discontinuity as either
removable or nonremovable.
x2 − 4x
f (x) =
55x 2 − x
4. Determine the value of a that makes the function continuous for all values of x.
3x + 7 x > a
ax + 1 x > 2
a. f (x) =  2
b. g(x) = 
x ≤2
5x − 1 x ≤ a
 x
€
5. Evaluate the following limit and write an "epsilon-delta" proof. lim ( 5x + 2 )
x→3
€
6. Find f ′(x) using the limit process.
a. f (x) = x
2
b. f (x) = x − 2x
7. Find all vertical asymptotes of each function:
x 2 + 5x
x
a. f (x) = 2
b. f (x) =
x − 25
sin x
MATH 1910
TEST 1 Review
8. Use the rules discussed in class to differentiate the following functions:
a. f (x) = 2x 5 − 8x + 3
d. y =
3
(2x )5
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5
7x 1 0
b. g(x) = 63 x
c. h(x) =
e. y = 3sin x
f. y = π 2 − 5cos x
9. Find all values of x for which each of the functions in parts a through d is not differentiable:
 3x + 4
x >0
a. f (x) = 
b. f (x) = x 4 / 3
4
+
3sin
x
x
≤
0

c.
d.
y
y
x
x
3
10. Let f (x) = x − 3x + 1.
a. Find an equation of the tangent line at (5,111).
b. For which values of x does the graph of f have a horizontal tangent line?
c. For which values of x does the graph of f have a tangent line parallel to graph of y = 7x + 1?
11. Sketch the graph of the derivative of each function graphed in 9c and 9d.
2
12. Consider the function f (x) = x + x .
a. Find the average rate of change in f over the interval [-1,2].
b. Find the instantaneous rate of change in f at x = 2.
13. An object dropped off the top of an 800-foot building is s feet above the ground t seconds after being
2
released, where s(t) = −16t + 800.
a. Find the velocity function.
b. Find the velocity of the object 2 seconds after release.
c. How long does it take the object to hit the ground?
d. What is the velocity of the object as it strikes the ground?
e. What is the average velocity of the object between t = 1 and t = 3?