Suppose we have a functionf(Q) =Q2+Q+ 2,Q0. Use calculus to find wheref(Q)/Qhas a minimum. Findh(x) for the following functions:

1. Suppose we have a function f(Q) = Q2 + Q + 2,Q ≥ 0. Use calculus to find where f(Q)/Q has a minimum.
2. Find h′(x) for the following functions:

a) h(x) = ex^2 + 1/x

(b) h(x) = (x^2 - x) (5x^5 + x^2)

(c) h(x) = 1/x-2

(d) h(x) = 1 + x/ x - 2. Draw a graph of h(x)

(e) h(x) = 1/ ax^2 + bx + c. What do you have to be careful about?

(f) h(x) = x^q e^px, where p and q are parameters. Find h′′(x) as well.

(g) h(x) = e^e^x

(h) h(x) = x^x.

(i) h(x) = x^a.

(j) h(x)=a^x. Assume a̸=e and a̸=0.

3) Suppose that P (Q) = a + bQ is a downward sloping linear demand curve. Define the revenue function as R(Q) = P(Q).Q. Use calculus to findR′(Q) and find the value of Q where R′(Q) = 0. Show what this means in a graph. What is the greatest value that the revenue function can take?

4) Suppose a firm produces an output measured in units Q. The cost of

producing Q units is given by the cost function C(Q) = aQ2 + bQ + c,

where you can assume a > 0,b > 0,c > 0. In Economics we also think

about cost per unit (average cost) given by: AC(Q) = C(Q) . EconomistsQ

also estimate the cost of producing 1 more unit of output by what we call marginal cost given by: MC(Q) = C′(Q) . Use calculus to help you graph both the AC(Q) and MC(Q) functions in one diagram and show that theMC(Q) function intersects the AC(Q) function from below at the point where AC(Q) is at its minimum.

5) Find d^3 y/ dx^3 if y = 1nx

6) y =f(x) = (1+x^2)^1/2. Find f′ and f′′.

7) y = f(x) = 120x - x^3/3 Is x∗ = 0 a maximum, minimum

or point of inflection?

8) Find Df for each of the following functions. Find and classify all critical

points of each function y = f(x).

(a) f(x) = x^3 e^x

(b) f(x) = x^2 2^x^2

(c) f(x) = 1n(4 - x^2)

(d) f(x) = (e^x - 1) (e^x - 4)

9) Suppose y(x) is implicitly defined by

3xe^xy^2 - 2y - 3x^2 - y^2 = 0

Approximate y(x) to first order around (x = 1, y = 0).

10) Consider the function y = -2.05 + 1.06x - 0.04x^2. Find all the critical

points of y and use the first and second derivative tests to classify them.

11) Consider the function y = x^2 e^-x which is defined over the interval [0,4]. Find all critical points and endpoints and classify them as (local or global) maxima, minima or points of inflection.

12) Suppose that f(x) is a concave function and g(x) is a function that is both increasing and concave. See if you can show that the composite functiony(x) = f(g(x)) is also concave.

13) Find the slope of the function f(x) = x^1/2, x > 0.

14) Consider the macroeconomic model:

Y = C + 1

C = A + B √y

Here: Y is national income, I is fixed investment that does not depend upon C or Y,and a > 0 and b > 0 are parameters. C stands for total consumption by all agents in the economy and, as can be seen from the second equation above, depends upon the level of national income.

(a) Solve for Y . This is called the equilibrium level of income in the

economy

Hint: It might help if you define Z = √Y so that Z^2 = Y. You should end up with a quadratic equation to solve.

Please explain and show step by step how every answer comes about. Thank you very much. Your help is greatly appreciated. This is not for a assignment or anything it is just practice for a test and it would be greatly appreciated if explained properly how to do each question so i could compare them to my answers. Thank you again.