S15
MATH 1225 – Test 1
23 Feb 2015
NAME:
NUMERIC ANS
CRN:
Use only methods from class. You must show work to receive credit.
1. (9 pts) Use the graph below of f (x) to estimate the following limits:
(a) lim f (x)
x→0+
2
(b) lim f (x)
(c) lim f (x)
x→3−
x→−3
3
DNE
2. (8 pts) The following is a graph of f (x).
(a) (3 pts) For ε = 3, mark on the portion of the graph where |f (x) − 8| < ε.
y interval (5, 11) or x interval (1.7, 2.22)
(b) (5 pts) For ε = 3, determine the maximum delta so that if 0 < |x − 2| < δ then |f (x) − 8| < ε.
About 0.22
3. (18 pts) Find the following limits using algebraic methods or theorems from class. Write ∞, −∞, and “Does not
exist” if appropriate. Justify your answer fully. Tables will not suffice.
x2 + 3x + 2
(a) (6 pts) lim √
x→−2
x2 + 5 − 3
3
2
x3 + 4x2 + 3
x→∞ 4x3 + 15x + 8
(b) (6 pts) lim
1
4
ex + cos(x)
x→0
x2 + 2
(c) (6 pts) lim
1
4. (12 pts) Define a piecewise function on [−1, 1] by
f (x) =
 2ax

+ b if x < 0


 sin(3x)
1




a − bx
if x = 0
if x > 0
Find a, b such that f (x) is continuous at x = 0.
a = 1, b =
1
3
5. (10 pts) Use a theorem from the class to prove that the equation e2x − 10x + 1 = 0 has a real root. (hint: e2 ≈ 7.4)
No final numeric answer.
6. (12 pts) Find the derivatives of the following functions:
cos(x3 )
+ 43x .
x2
2 cos(x3 )
−3 sin(x3 ) −
+ 3 ln(4)43x
x3
(b) (5 pts) f (x) = 3 tan(x) + x4 ex .
(a) (7 pts) f (x) =
3 sec2 (x) + ex x3 (x + 4)
7. (16 pts) For 0 ≤ t ≤ 4 the height of a firework t seconds after its start can be described by h(t) = 9t2 − 2t3 .
(a) (6 pts) Find the average velocity on each of the intervals [0, 2], [2, 3], [3, 4].
10, 7, -11 units/s respectively
1
(b) (4 pts) When is the firework’s velocity zero?
t = 0 and t = 3
(c) (6 pts) Find the total distance traveled between t = 0 and t = 4.
38 units
8. (15 pts) Consider the function f (x) =
1
.
x+3
(a) (9 pts) Use the formal definition of the derivative to find f 0 (x). (No credit will be given for using derivative
rules.)
f 0 (x) =
−1
(x+3)2
(b) (2 pts) What is the domain of f 0 (x)?
x 6= −3
(c) (4 pts) Find the tangent line to f at x = 2.
1
x+
y = − 25
7
25
2