NAME: Brief Answers
TEST3M/MAC2311
Page 1 of 5
Read Me First:
Show all essential work very neatly. Use
correct notation when presenting your computations and arguments.
Write using complete sentences. Be careful.
Remember this: "="
denotes "equals" , "
⇒
" denotes "implies" , and "
⇔
" denotes "is
equivalent to".
Do not "box" your answers. Communicate. Show
me all the magic on the page.
1. (10 pts.)
(a) Using implicit differentiation, compute dy/dx
and d
2
y/dx
2
when x
2
+ y
2
= 25.
Label your expressions correctly
or else.
d(x
2
+ y
2
)/dx = d(25)/dx implies that dy/dx = x/y.
Consequently, after differentiating one more time using quotient
rule, replacing the occurance of dy/dx in the expression for the
second derivative, and cleaning up the algebra, we obtain
d
2
y/dx
2
= [y
2
+ x
2
]/y
3
.
(b) Obtain an equation for the line tangent to the graph of
x
2
+ y
2
= 25 at the point (1,(24)
1/2
).
Evaluating the implicit derivative above at (1,(24)
1/2
) provides
us with the slope of the tangent line, namely 1/(24)
1/2
.
Consequently, an equation for the tangent line at the desired
point is y ((24)
1/2
) = 1/(24)
1/2
(x  1).
2. (5 pts.)
A 5ft. ladder is leaning against the wall. If
the top of the ladder slips down the wall at a rate of 2
ft./sec., how fast will the foot be moving away from the wall
when the top is 3 ft. above the ground? [Try 4.6: 1215 1st!]
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 Fall '08
 STAFF
 Calculus, Derivative, lim, Logarithm, 2 feet, 2 ft, 0.5 feet

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