Exam # 1
Fall 2005
MATH 1100-05
Instructor: Oana Veliche
Time: 1 hour
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) There are 10 problems, each worth 10 points.
(3) Justify all you answers. Correct answers with no justification will not be given any credit.
(4) No books, notes or calculators may be used.
(5) A list of definitions and formulas is attached at the end.
Problem #
1
2
3
4
5
# Points
1
6
7
8
9
10
Total
2
Problem 1. Find the inverse f −1 , of the function
√
f (x) = 4 − 2x, x ≤ 2.
(Write the domain of the inverse function as well.)
3
Problem 2. Consider the function f (x) =
(a) Find lim f (x) =
x2
x−3
.
− 2x − 3
x→1
(b) Find lim f (x) =
x→3
(c) Find all the points of discontinuity of f . Which one of the continuity conditions fails? (See
the definitions and formulas page.)
(d) Describe the intervals on which the function is continuous.
4
Problem 3. Consider the function f (x) = x3 − 3x.
(a) Find the critical points.
(b) Using the table below determine the intervals on which the function is increasing or decreasing.
x
f ′ (x)
f (x)
(c) Sketch the graph of the function.
5
Problem 4. Using the limit definition, find the derivative of f (x) = 1 − x2 .
6
Problem 5. Consider the function
√
3
f (x) = x x + √
.
3
x
(a) Find f ′ (x).
(b) Find the slope of the tangent line to the graph of the function f at the point (1, 4).
(c) Find the equation of the tangent line to the graph of the function f at the point (1, 4).
7
Problem 6. If the revenue obtained from selling x units is given by the function
R(x) = x(1000 + 200x).
Find the following:
(a) The average rate of change of the revenue on the interval [5, 10].
(b) The marginal revenue for a production level of 10 units.
8
Problem 7. Find the derivatives of the following functions:
(a) f (x) = (3 + x2 ) · (1 − x), by using the product rule.
(b) g(x) =
3 − 2x
, by using the quotient rule.
1 + x3
9
Problem 8. Find the derivative of the function:
y = x3 + 4
Problem 9. Find f ′′′ (1) for the function f (x) =
1
.
x2
31
.
10
dy
, if
dx
x2 + x2 y − y 3 = 5.
Problem 10. Use implicit differentiation to find
11
Useful definitions and formulas
(1) Let c be a number in the interval (a, b). A function f is continuous at the point c if the
following conditions are true:
1. f (c) is defined.
2. lim f (x) exists.
x→c
3. lim f (x) = f (c).
x→c
f (x + ∆x) − f (x)
.
∆x→0
∆x
(2) Definition of the derivative: f ′ (x) = lim
(3) Average rate of change:
f (b) − f (a)
.
b−a
(4) The product rule: (f · g)′ = f ′ · g + f · g ′ .
′
f ′ · g − f · g′
f
.
=
(5) The quotient rule:
g
g2
(6) The simple power rule:
d n
[x ] = nxn−1 .
dx
(7) The generalized power rule: If u is a function of x, then
d n
[u ] = nun−1u′ .
dx