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TEST 1
Math 105 2/10/12 Name: (^) ︸ ︷︷ ︸ by writing my name I swear this work is my own
Read all of the following information before starting the exam:
- Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct).
- Circle or otherwise indicate your final answers.
- Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements. Put a smiley face next to your name for one point.
- This test has 6 problems and is worth 100 points, It is your responsibility to make sure that you have all of the pages!
- Good luck!
1. (10 points) The position of a car after time t is given by the table of values below.
t (seconds) 0 1 2 3 4 5 s(t) (feet) 0 15 46 72 118 195 a. (3 pts) Find the average velocity for the time period beginning when t = 2 and lasting 3 sec. s(5) − s(2) 3
= 49. 66 f t/sec
b. (3 pts) Find the average velocity for the time period beginning when t = 2 and lasting 1 sec. s(3) − s(2) 1 = 26f t/sec
c. (4 pts) Estimate the instantaneous velocity when t = 2. s(1) − s(2) − 1
= 31f t/sec.
Approximately,
- 5 f t/sec.
Or use the secant through [1,3]. You will get the same answer.
2. (10 points) Use the formal definition of the derivative (ie. f ′(x) = lim
h→ 0
f (x+h)−f (x) h ) to prove that d dx (cg(x)) =^ c^
d dx g(x) where^ c^ is a constant.
(cg(x))′^ = lim h→ 0
cg(x + h) − cg(x) h
= lim h→ 0 c
g(x + h) − g(x) h
= c lim h→ 0
g(x + h) − g(x) h
= cg′(x)
3. (19 points) The graph of f (x) is given. Solving the following (assume the tickmarks occur at 1, 2,
etc).
a. (3 pts) lim x→ 1 −
f (x) = − 1 b. (3 pts) lim x→ 1 +^
f (x) = 2 c. (3 pts) f (1) = − 1 d. (3 pts) lim x→− 2 f (x) = DN E e. (3 pts) lim x→− 3 f (x) = 3 f. (4 pts) For what values of x is f (x) NOT continuous? f (x) is NOT continuous at x = − 3 , − 2 , 1 , 3.
5. (12 points) Find the following limits. Use algebra when possible or necessary. If a limit doesn’t exist
then clearly explain why. a. (4 pts) lim x→ 1
7 x− 7 3 x^2 − 2 lim x→ 1
7 x − 7 3 x^2 − 2
b. (4 pts) lim x→ 0
√ 25 −x− 5 x
lim x→ 0
25 − x − 5 x
25 − x + 5 √ 25 − x + 5
= lim x→ 0
25 − x − 25 x(
25 − x + 5)
= lim x→ 0
x(
25 − x + 5)
c. (4 pts) lim x→ 3
|x− 3 | x− 3 Use a table of values to show that the limit doesn’t exist because the right hand and left hand limits are different. x |x x−−^33 | 2.99 - 2.999 - 3.001 1 3.01 1
6. (14 points)
a. (5 pts) Solve the differential equation y′^ = 4x^3 − (^) x^62 + 2
x.
y = x^4 +
x
x^3 + C
b. (5 pts) What is y′′? 12 x^2 +
x^3
x
. c. (4 pts) If you haven’t done so already, write your answers from a. and b. without fractional or negative exponents. See a. and b..
