Review of Mathematics
Function
1. What is a function? What is a dependent and
independent variable?
2. Draw graphs of the following functions (corre-
spondences):
(a) p = 2x − 5
(b) x2 = 5 −
√
x1
(c) x2 = 5 − x2
1
(d) x2 = 4/x1
(e) 10 = min{x1, x2}
(f) 10 = min{x1 + 2x2, 2x1 + x2}
(g) S(p) = s1(p) + s2(p),
where s1(p) = p a s2(p) = 2p.
(h) S(p) = s1(p) + s2(p),
where s1(p) = 2p a s2(p) = p − 1 for p ≥ 1
and s2(p) = 0 for p < 1.
3. What is a continuous function? What is the opposite
to the continuous function?
4. What is a monotonic function? What is the shape
of a positive and negative monotonic function?
5. What is the shape of a convex and concave
function?
6. What is the inverse function? Formulate the inverse
function of:
(a) y = ax + b
(b) y = 5/x
(c) y = ex
Logarithm
7. What is the logarithm? Draw the function
y = ln x.
8. Calculate the logarithm of xa
1xb
2.
Derivatives
9. What is the derivative? What is the relationship
between the derivative and the slope of a tangent
line to a function? What is the relationship between
derivatives and convexity (concavity) of a
function?
10. What is the product rule? What is the chain rule?
11. Take the derivative of the following functions
with respect to p:
(a) D(p) = 50 − 2p
(b) D(p) = 30p−2
(c) D(p) = (2p + a)(−b)
(d) R(p) = pq(p)
12. What is a partial derivative? Take a partial derivative
of the following functions with respect to
x1 a x2:
(a) U(x1, x2) = ax1 + bx2
(b) f(x1, x2) = xa
1xb
2
(c) U(x1, x2) = a ln x1 + bx2
(d) U(x1, x2) = a
√
x1 + bx2
(e) U(x1, x2) = (x2
1 + x2
2)a
Optimalization
13. Solve the following problem:
max
x1,x2
c ln x1 + d ln x2
subject to p1x1 + p2x2 = m,
where 0 < c < 1, 0 < d < 1, p1 > 0, p2 > 0
a m > 0 are constants.