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(1) Calculate
pthe following derivatives:
d 4
(a) dx
( sin(cos x)) Ignore this problem!
√
(b)
d
2
dx (x e
(c) xx
(x−1)(x−2)(x−3)
2
(d) log10 (100x2 )
).
2
(2) Write the equation for the line tangent to y 2 sin(2x) = −2y at the point (π/4, −2). Ignore
this! We didn’t talk about trig functions!
Math 148.
3
(3) (a) Demand is governed by q = p−1/2 ln p. Find η for p = 10.
(b) Suppose p = 20, q = 10, and
increases by 5%.
dp
dq
= −2. Estimate the change in demand if price
4
√
(4) Demand is governed by the equation q = ep , and supply is goverened by q = 10 p. Thus,
√
the equilibrium price is a solution to the quation ep = 10 p. This equation has a solution
somewhere in the interval between 2 and 3. Estimate the solution using two repetitions of
Newton’s Method.
Math 148.
(5) Where is the following function concave up and concave down?
1 4
f (x) = − 10
x + 43 x2 .
5
6
(6) Find any local minima and maxima of the function f (x) = ex /x. Does f have an absolute
minimum or absolute maximum for x > 0?
Math 148.
7
(7) The relationship between price and demand for a product is p = 52 q 2 − 2q + 10. Estimate,
using differentials, how much price will change if demand goes up by 0.1.
8
(8) If f is some differentiable function, f (5) = 2 and f 0 (5) = 3, approximate f (5.05).
Math 148.
R
(9) Find (ex − 4x3 + x1 )dx.
9
10
(10) If marginal cost is M C(q) = q 2 + q + 1 and the total cost to produce 3 units is 10, what
are fixed costs?