(Will not be accepted without staple) [First Name] [Last Name] Math-2417, thq 02, 26-08-16, 10:51 [Net ID] [Problem Section] Instructions:
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2. This handout is due at the beginning of problem section.
3. This handout must be printed out and stapled. You may print it
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4. Your work must be hand written on this handout.
5. You must show all work. You may receive zero or reduced points
for insufficient work.
6. Your work must be neatly organized and written. You may receive
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7. Only a subset of these questions will be graded. You will not be
told which questions will be graded in advance. Page 2 of 10 Math-2417, thq 02, 26-08-16, 10:51 Questions 1 2 3 4 5 6 7 8 9 Total Points 10 10 10 10 10 10 10 10 10 90 Score
1. (10 points) Find the limit if it exists. If it doesn’t exist, explain why.
√
x−3
(a) lim
x→3+ x − 9 (b) lim x−5
|x − 5| (c) lim (x − h)2 + x − h − x2 − x
h x→5+ h→0− Page 3 of 10 Math-2417, thq 02, 26-08-16, 10:51 2. (10 points) Find the x-values (if any) at which f is not continuous. Which of the discontinuities are
removable?
(a) f (x) = 5
3−x (b) f (x) = sec(x) (c) f (x) = 3 − x, x < −1 x2 , x ≥ −1 Page 4 of 10 Math-2417, thq 02, 26-08-16, 10:51 3. (10 points) Find value(s) of parameter c (if exists) that make f continuous everywhere.
(a) f (x) = 2
x+c (b) f (x) = 2
x2 + 4x + c Page 5 of 10 Math-2417, thq 02, 26-08-16, 10:51 4. (10 points) Does the function has a zero in the given interval? Explain why.
(a) f (x) = x4 − 5 on [−1, 2] (b) f (x) = tan(2x) − 4 on [−π, π] Page 6 of 10 Math-2417, thq 02, 26-08-16, 10:51 5. (10 points) Let f (x) = 3x2 + ax + b.
Find values of the parameters a and b so that the following equations are true.
(i) lim f (x) = 12
x→2 (ii) lim f (x) = 6
x→−1 and Page 7 of 10 Math-2417, thq 02, 26-08-16, 10:51 6. (10 points) Find all values of the parameters a and b so that the following function has vertical asymptotes
at x = 2 and x = −3.
x+7
f (x) = 2
x + ax + b Page 8 of 10
7. (10 points) Find all vertical asymptotes of the following function.
√
√
x − 3 x2 + 4
f (x) =
(x + 4)(x − 3)(x − 5) Math-2417, thq 02, 26-08-16, 10:51 Page 9 of 10 Math-2417, thq 02, 26-08-16, 10:51 8. (10 points) Compute the derivative of the following function using the limit definition of derivative.
1
g(x) = x −
x Page 10 of 10 Math-2417, thq 02, 26-08-16, 10:51 9. (10 points) Consider the following function. x, x ≤ 0
f (x) = x2 , x > 0
Away from the join point x = 0, it is clear that f (x) is differentiable because x and x2 are polynomials
and polynomials are differentiable. Is this function differentiable at x = 0, i.e., did we “glue” these
polynomials together “smoothly”? Use the definition of derivative to show that f (x) is not differentiable
at x = 0. To do this, you must show the limit as ∆x → 0− (from the left) is different from the limit as
∆x → 0+ (from the right) in the definition of derivative.