Math 113 Hw 5 Worksheet With Answers Page 4
ADVERTISEMENT
1
2
3
4
59. Exercise 2.7.32. The limit
16 + h
2
lim
h
0
represents the derivative of some function f at some number a. State such an f and a.
Answer: If we let f (x) =
16 + x and choose a = 0, then
16 + (0 + h)
16 + 0
f (0 + h)
f (0)
f (a) = lim
= lim
h
h
0
0
16 + h
16
= lim
h
0
16 + h
2
= lim
,
h
0
which is the given limit. Therefore, this limit represents f (0) for f (x) =
16 + x.
10. Exercise 2.7.48. The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee
company at a price of p dollars per pound is Q = f (p).
(a) What is the meaning of the derivative f (8)? What are its units?
Answer: The derivative of a function always gives the change in the output with respect
to the change in the input. In this case, since the input to the function is the price and
the output is the quantity sold, the derivative gives the change in quantity sold with
respect to a change in the price. This is essentially what economists call the “price
elasticity of demand”.
In particular, f (8) will give the expected change in quantity sold if the price is changed
from $8/lb. The units are pounds per dollar.
(b) Is f (8) positive or negative? Explain.
Answer: f (8) will almost certainly be negative. Remember that
f (8 + h)
f (8)
f (8) = lim
.
h
0
If h > 0, then f (8 + h) represents the quantity of coffee sold for some price that’s greater
than $8/lb., which we would certainly expect to be less than f (8) (the quantity sold
when the price is exactly $8/lb.). Hence, the number f (8 + h)
f (8) will be negative
and so, since h > 0,
f (8 + h)
f (8)
< 0.
h
On the other hand, when h < 0, the numerator inside the above limit will be the
difference between the quantity sold when the price is less than $8/lb. and the quantity
sold when the price is exactly $8/lb. This will almost certainly be a positive number, so
again (since h < 0) we’ll have
f (8 + h)
f (8)
< 0.
h
Hence, the expression inside the limit will always be negative, so we would expect f (8)
to be negative.
4
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education