Mat 210, Spring 2001 – Ela Jackiewicz
ã 2001 Arizona State University Department of Mathematics
Exam 1 Key – Covers Chapters 2 and 3
1. Find the following limits. If the limit does not exist clearly explain why.
[4 pts each]
a. = 8 (since e^(-x) -> 0 as x ->+ infinity, check on the graph)
b. = - 17/3
c. =4 ( factor numerator as (x-2)(x+2) and simplify)
d.= 1/4( multiply numerator and denominator by (1/x^3),
expressions like 2/x^2, 5/x^3 will approach 0 as x -> +infinity
e. =
2. Consider the function .
a. Using the limit definition, find the instantaneous rate of change of at [9pts]
(1/h)[f(3+h)-f(3)]=(1/h) [(3+h)^2-4(3+h) – (3^2-4*3)]= (1/h)[ 9+6h+h^2-12-4h-9+12]
=(1/h)[h^2+2h]=h+2
limit(h+2)=2(as h->0)
b. Using your answer to part a) find the equation of the tangent line at the point [6pts]
f(3)= - 3, from part a) slope = 2, y+3=2(x-3) gives y=2x-9
3. The graph of is given below.
b) Find . If the limit does not exist clearly explain why. [3pts]
does not exist, since left limit(1) is not the same as right limit (3)
a) Find. If the limit does not exist clearly explain why. [3pts]
=4
4. Fill in the table and then estimate
if it exists. [5pts]
x / .9 / .99 / .999 / 1.001 / 1.01 / 1.1/ .2372 / .2487 / .2498 / .2501 / .2512 / .2623
=.25
5. Find the derivative of the following functions. [5pts each]
a.
b.
c.
d.
6. A particle moves according to the law . Find the velocity at [10pts]
V(t)=s’(t)= v(5) = (68/729)*(e^5) ~ 13.84
7. The profit function on a certain commodity is given by
, where is the number of items produced
and is the cost in hundred of dollars to produce items.
a. Find the marginal profit function. [4pts]
P’(x)=1.8x^2-6x+12
b. For find the approximation for using the marginal profit function from part (a.) [3pts]
P’(20) = 612
c. For , find the profit on the next item, Compare the result to part (b). [3pts]
P(21)-P(20)=645.6
Numbers are relatively close.
8. Find the points on the graph of where the tangent line to the graph is horizontal. [10pts]
f’(x)= e^x(2x+x^2)=0
x=0 or x=-2
f(0)=0 f(-2)=4e^(-2)
(0,0) and (-2, .54) are the points