1
Math1013-L2/L6 Calculus I
Some Basic Problems on Applications of Derivatives: Extreme Values, Mean
Value Theorem and the Shape of a Graph
Finding maximum and minimum of a continuous function on a closed interval: A matter of checking function
values at
Math1013-L2/L6 Calculus I
Brief Answers to Week 1 Class Problem Set
1. A taxi company charges two dollars for the rst kilometer (or part of a kilometer) and 20 cents for
each succeeding tenth of a kilometer (or part). Consider the cost C (in dollars) of a
Math1013-L2/L6 Calculus I
Week 1: Brief Review of Functions
Before proceeding into calculus, you should get familiar with all basic notations and concepts about functions.
A function f is just a rule to assign to each x in a set of numbers called domain
1
Math1013-L2/L6 Calculus I
Basic Problems on Inverse Functions
Things to review:
What is a one-to-one function?
What is the relationship between f and its inverse f 1 (if exists), and their graphs?
If you really understand y = ax , you should also kno
1
Math1013-L2/L6 Calculus I
Basic Problems on Derivatives
Get use to the limit denition of derivative: (i.e., by looking at slopes of nearby secant lines)
f (x) = lim
h0
f (x + x) f (x) or
f (t) f (x) or
y
f (x + h) f (x) or
= lim
= lim
= lim
t x
x 0
x
1
Math1013 L2/L6 Calculus I
More Basic Problems on Areas, Denite Integrals, Fundamental Theorem of
Calculus, and the Substitution Rule
Approximate areas using rectangles.
b
Understand the meaning of the denite integral
rectangular areas).
f (x)dx as the
1
Math1013-L2/L6 Calculus I
Basic Problems on Derivatives
Get use to the limit denition of derivative: (i.e., by looking at slopes of nearby secant lines)
f (x) = lim
h0
f (x + h) f (x) or
f (x + x) f (x) or
f (t) f (x) or
y
= lim
= lim
= lim
t x
x 0
x
1
Math1013 L2/L6 Calculus I
More Basic Problems on Areas, Denite Integrals, Fundamental Theorem of
Calculus, and the Substitution Rule
Approximate areas using rectangles.
b
Understand the meaning of the denite integral
f (x)dx as the limit of Riemann su
1
Math1013-L2/L6 Calculus I
Some Basic Problems on Applications of Derivatives: Extreme Values, Mean
Value Theorem and the Shape of a Graph
1. Find the absolute maximum and absolute minimum values of f on the given interval.
(a) f (x) =
(x2
x
, [2, 2].
+
1
Math1013 Calculus I
Brief Answers to Class Problem Set
Class Problem Set: Using Derivatives: LHopitals Rule, Max-Min Problems, Antiderivatives
1. Find the limit:
e x e x 2 x
(a) lim
x0
x sin x
(d)
1x2x
3 + 2
3
3
lim+
x0
(a) It is a
lim
x 0
(b)
0
0
1
x
l