Definition of Derivative at a Point
Use the definition of derivative at a point to evaluate the derivative of each of the following functions at
the given point.
1. f (x) = 2x 1 at x = 1
SOLUTION: 2
2. f (x) = 5x2 + x 7 at x = 3
SOLUTION: 31
3. f (x) =
3x
Piecewise Functions, Limits, and Continuity
(1) Sketch the piecewise function below, and then
1
x2
2
f (x) = 3
x + 1
1
x
(a)
lim f (x) =
x1+
(b) lim f (x) =
x1+
(c) lim f (x) =
x2+
(d) lim f (x) =
x3
find the requested limits.
if
if
if
if
if
x < 1,
1 x
Graphs and Limits
(1) Draw the graph of a function g whose domain is the set of all real numbers and where:
(a) lim g(x) = 1
x
(b)
lim g(x) =
x
(c) lim g(x) =
x1
(d) g(3) = 0
(e) lim g(x) = 0
x0
(f) g(0) = 2
(g) lim g(x) = 1
x1
(h) lim g(x) = 1
x1+
(2)
Trigonometry
(1) Sketch the graphs of all six trig functions.
(2) Be able to complete the trig table from class by yourself! (Dont memorize. Thats really not helpful.
Instead, be able to sketch a quick unit circle and derive the trig values.)
(3) Find the
Using Alebra to Find Limits
(1) Find the following limits:
x1
(a) lim 3
x1 x 1
v2 4
v2 v 3 + 8
(b) lim
(c) lim
1
2
2
x1 x 1
(d) lim
1
1
t
t 1+t
x1
t0
3x2 + ax + a + 3
exists as a finite number?
x2
x2 + x 2
(2) Is there a number a such that lim
Asymptote Homework
1.
Give a rational expression that has vertical asymptotes at x = 3 and x = -2, and a horizontal
asymptote at y = 4.
2. Give a rational expression that has vertical asymptotes at x = 0 and x = -2, and a horizontal
asymptote at y = 0.
3.
CHAPTER 3 DERIVATIVES
A graphing calculator or computer provides another way of looking at differentiabil _
If f is differentiable at a. then when we zoom in toward the point (a. f (al) the g .
strai-ghtens DUI and appears more and more like a line. (Se
2. Find the dimensions of the rectangle of largest area having xed perimeter 100. =>
4. A box with square base and no top is to hold a volume 100. Find the dimensions of the box
that requires the least material for the ve sides. Also nd the ratio of heigh