# What is CALCULUS? - An Introduction

Contributed by:
OUTLINE:
1. Review of functions and graphs
2. Review of limits
3. Review of derivatives - the idea of velocity, tangent, and normal lines to curves
4. Review of related rates and max/min problems
1. to MATH 104-002:
Calculus I
2. Welcome to the Course
1. Math 104 – Calculus I
2. Topics: quick review of Math 103 topics, methods and
applications of integration, infinite series and applications.
Moves very fast
Demanding workload, but help is available!
YOU ARE ADULTS - how much do you need to practice
each topic?
Emphasis on applications - what is this stuff good for?
4. Opportunities to interact with professor, TA, and other
3. Outline for Week 1
(a)Review of functions and graphs
(b)Review of limits
(c)Review of derivatives - idea of velocity,
tangent and normal lines to curves
(d)Review of related rates and max/min
problems
4. Functions and Graphs
The idea of a function and of the graph of a function should be very familiar
4 3 2
f (x) = 6x − x − 9x + 4 x
5. Questions for discussion...
1. Describe the graph of the function f(x) (use calculus vocabulary as
appropriate).
2. The graph intersects the y-axis at one point. What is it (how do
you find it)?
3. How do you know there are no other points where the graph
intersects the y-axis?
4. The graph intersects the x-axis at four points. What are they (how
do you find them)?
5. How do you know there are no other points where the graph
intersects the x-axis?
6. The graph has a low point around x=4, y=-100. What is it exactly?
How do you find it?
7. Where might this function come from?
6. Kinds of functions that
should be familiar:
Polynomials, quotients of polynomials
Powers and roots
Exponential, logarithmic
Trigonometric functions (sine, cosine, tangent, secant,
cotangent, cosecant)
Hyperbolic functions (sinh, cosh, tanh, sech, coth, csch)
7. Quick Question
The domain of the function
1 x is...
f(x)
2
x  2x
A. All x except x=0, x=2
B. All x < 1 except x=0.
C. All x > 1 except x=2.
D. All x < 1.
E. All x > 1.
8. Quick Question
Which of the following has a graph that
is symmetric with respect to the y-axis?
A. y=
x  1 5 2
D. y= x  2x
x
x
4 E. y= 3
B. y= 2x  1 x 3
3
C. y= x  2x
9. Quick Question
The period of the function
 3 x 
f ( x) sin   is...
 5 
A. 3
B. 3/5
C. 10/3
D. 6/5
E. 5
10. Quick Question
a
 a
If log a 5  , then a=
3
A. 5
B. 15
C. 25
D. 125
E. None of these
11. Limits
The concept of limit underlies all of calculus.
Derivatives, integrals and series are all different kinds of limits.
Limits are one way that mathematicians deal with the infinite.
12. First things first...
First some notation and a few basic facts.
Let f be a function, and let a and L be fixed numbers.
Then lim f( x) L is read
x a
"the limit of f(x) as x approaches a is L"
You probably have an intuitive idea of what this means.
And we can do examples:
13. For many functions...
...and many values of a , it is true that
lim f ( x) f (a )
x a
And it is usually apparent when this is not true.
"Interesting" things happen when f(a) is not
well-defined, or there is something "singular"
14. Definition of Limit
So it is generally pretty clear what we
mean by
lim f ( x) L
x a
But what is the formal mathematical
15. Properties of real numbers
One of the reasons that limits are so difficult to
define is that a limit, if it exists, is a real number.
And it is hard to define precisely what is meant
by the system of real numbers.
Besides algebraic and order properties (which
also pertain to the system of rational numbers),
the real numbers have a continuity property.
16. Least upper bound property
If a set of real numbers has
an upper bound, then it has
a least upper bound.
17. Important example
2
The set of real numbers x such that x  2. The
corresponding set of rational numbers has no least
upper bound. But the set of reals has the number 2
In an Advanced Calculus course, you learn how to start
from this property and construct the system of real
numbers, and how the definition of limit works from
18. Official definition
lim f(x) = L means that for any ε > 0,
x→ a
no matter how small, you can find a δ > 0
such that if x is within δ of a, i.e., if x-a < δ,
then f(x)-L < ε
19. For example….
2
lim x 25
x 5
because if   1 and we choose   111 
Then for all x such that x  5   we have
5    x  5   and so
25  10   2  x 2  25  10   2
which implies
2
2  
2
x  25  10    10  
11 121
20. Top ten famous limits:
1 1
1. lim  lim 
x 0 x x 0 x
1
2. lim 0
x  x
21. n
3. (A) If 0 < x < 1 then lim x 0
n 
n
(B) If x > 1, then lim x 
n 
sin x 1  cos x
4. lim 1 and lim 0
x 0 x x 0 x
x x
5. lim e 0 and lim e 
x   x 
22. n
x
6. For any value of n, lim x 0
x  e
ln x
and for any positive value of n, lim n 0
x  x
1
lim sin   6-10
7. x 0
 x
does not exist!
23. 8. lim x ln( x) 0
x 0
x
 1
9. lim  1   e
x 
 x
10. If f is differentiable at a, then
f ( x)  f (a)
lim  f ' (a)
x a x a
24. Basic properties of
limits
I. Arithmetic of limits:
If both lim f ( x) and lim g ( x) exist, then
x a x a
lim f ( x) g ( x) lim f ( x) lim g ( x)
x a x a x a
lim f ( x) g ( x) lim f ( x) lim g ( x)
x a x a x a
and if f ( x) lim f ( x)
lim g ( x) 0 , then lim  x a
x a x a g ( x) lim g ( x)
x a
25. II. Two-sided and one-sided
limits:
lim f ( x) L if and only if
x a
BOTH lim f ( x) L and lim f ( x) L
x a  x a 
III. Monotonicity:
If f(x) g(x) for all x near a,
then lim f( x) lim g( x)
x a x a
26. IV. Squeeze theorem:
If f(x)  g(x) h(x) for all x near a, and if
lim f ( x) lim h( x), then lim g( x) exists and
x a x a x a
is equal to the common value of the other
two limits.
1
−x ≤x sin ≤x
x
1
lim x sin = 0
x→ 0 x
27. Let’s work through a few:
x 5 x 5
lim lim
x 2 x  2 x  2 x  2
2
x 4
lim
x  2 x  2
28. Now you try this one...
2 t  2
lim 
t 0 t
A. 0 E. -1
B.  F. 2
C. -1/2 G. -2
1 1
D. H. 
2 2 2 2
29. Continuity
A function f is continuous at x = a if it is true
that lim
x→ a
f(x) = f(a)
(The existence of both the limit and of f(a) is
implicit here).
€ Functions that are continuous at every point
of an interval are called "continuous on the
interval".
30. Intermediate value theorem
The most important property of continuous functions is the
"common sense" Intermediate Value Theorem:
Suppose f is continuous on the interval [a,b], and f(a) = m, and
f(b) = M, with m < M. Then for any number p between m and
M, there is a solution in [a,b] of the equation f(x) = p.
31. Application of the intermediate-value theorem
3
f ( x)  x  2 x  2
Maple graph
Since f(0)=-2 and f(2)=+2, there must be a root of f(x)=0 in
between x=0 and x=2. A naive way to look for it is the
"bisection method" -- try the number halfway between the
two closest places you know of where f has opposite signs.
32. 3
f ( x)  x  2 x  2
We know that f(0) = -2 and f(2) = 2, so there is a root
in between. Choose the halfway point, x = 1.
Since f(1) = -3 < 0, we now know (of course, we already
knew from the graph) that there is a root between 1 and 2.
So try halfway between again:
f(1.5) = -1.625
So the root is between 1.5 and 2. Try 1.75:
f(1.75) = -.140625
33. 3
f ( x) x  2 x  2
We had f(1.75) < 0 and f(2) > 0. So the root is
between 1.75 and 2. Try the average, x = 1.875
f(1.875) = .841796875
f is positive here, so the root is between 1.75 and 1.875.
Try their average (x=1.8125):
f(1.8125) = .329345703
So the root is between 1.75 and 1.8125. One more:
f (1.78125) = .089141846
So now we know the root is between 1.75 and 1.8125.
You could write a computer program to continue this to
any desired accuracy.
34. Derivatives
Let’s discuss it:
1. What, in a few words, is the derivative of a function?
2. What are some things you learn about the graph of a
function from its derivative?
3. What are some applications of the derivative?
4. What is a differential? What does dy = f '(x) dx mean?
35. Derivatives (continued)
Derivatives give a comparison between the rates of
change of two variables:
When x changes by so much, then y changes by so much.
Derivatives are like "exchange rates".
6/03/10 1 US Dollar = 0.83 Euro
1 Euro = 1.204 US Dollar (USD)
6/04/10 1 US Dollar = 0.85 Euro
1 Euro = 1.176 US Dollar (USD)
Definition of derivative:
dy f ( x  h)  f ( x )
lim
dx h 0 h
36. Common derivative formulas:
d
d p
 
x  px p  1  f ( x) g ( x)   f ( x) dg  df g ( x)
dx dx dx dx
d x
dx
 
e e x d  f ( x)  g ( x) f ' ( x)  f ( x) g ' ( x)
  
dx  g ( x)   g ( x)  2
d 1
 ln x   d
dx x  f ( g ( x)   f ' ( g ( x)) g ' ( x)
dx
d
 sin x  cos x
dx
d Let’s do some examples…..
 cos x   sin x
37. Derivative question #1
5 1
Find f '(1) if f ( x)  x 
x9 / 5
A. 1/5 E. -1/5
B. 2/5 F. 4/5
C. -8/5 G. 8/5
D. -2/5 H. -4/5
38. Derivative question #2
8
Find the equation of a line tangent to y
4  3x
at the point (4,2).
A. 6x+y=26 E. 5x+21y=62
B. 4x+2y=20 F. 4x+15y=46
C. 3x-4y=4 G. 3x+16y=44
D. 7x+18y=64 H. 2x-y=6
39. Derivative question #3
x
2 e
Calculate d f if f ( x) 
dx 2 x
x
e  x  4 4 e x  x  2
A. E.
x x3

ex x2  1  4 F.

e x x 2  5x 
x x 3
x

e x x 2
 G.

ex x2  2x  2  3
x 4
x
x
e x 3  2
 
e x x3  4 x 2  3  3
D. 4 H. x
x
40. Derivative question #4
What is the largest interval on which the
x
function f ( x)  2 is concave upward?
x 1
A. (0,1) E. (1, 3 )
B. (1,2) F. ( 3 ,  )
C. (1, ) G. ( 2 , )
D. (0, ) H. (1/2,  )
41. Discussion
Here is the graph of a function.
Draw a graph of its derivative.
42. The meaning and uses of
derivatives, in particular:
• (a) The idea of linear approximation
• (b) How second derivatives are related to
• (c) Together, these two ideas help to solve
max/min problems
43. Basic functions --linear and
• The derivative and second derivative
provide us with a way of comparing
other functions with (and approximating
them by) linear and quadratic functions.
• Before you can do that, though, you need
functions.
44. Let’s review
• Let's review: linear functions of
one variable in the plane are
determined by one point + slope
(one number):
• y = 4 + 3(x-2)
45. Linear functions
• Linear functions occur in calculus as
differential approximations to more
complicated functions (or first-order
Taylor polynomials):
• f(x) = f(a) + f '(a) (x-a)
(approximately)
• Quadratic functions have parabolas as
their graphs:
2 2
x x
y   x  2, y   x 1
2 2
• Quadratic functions occur as second-
order Taylor polynomials:
• f(x) = f(a) + f '(a)(x-a) + f "(a)(x-a)2/2!
(approximately)
48. They also help us tell...
• … relative maximums from relative
minimums -- if f '(a) =0 the quadratic
approximation reduces to
• f(x) = f(a) + f "(a)(x-a)2/2! and the
sign of f "(a) tells us whether x=a is a
relative max (f "(a)<0) or a relative min (f
"(a)>0).
49. by way of -review,
max recall
and that
minto problems
find the
maximum and minimum values of a function on any
interval, we should look at three kinds of points:
1. The critical points of the function. These are the points where
the derivative of the function is equal to zero.
2. The places where the derivative of the function fails to exist
(sometimes these are called critical points,too).
3. The endpoints of the interval. If the interval is unbounded,
this means paying attention to
lim f ( x) and/or lim f ( x).
x  x  (  )
50. Position, velocity, and acceleration:
You know that if y = f(t) represents the position of an object
moving along a line, the v = f '(t) is its velocity, and a = f "(t) is
its acceleration.
2
Example: For falling objects, y = 0 y  v0 t  16t
is the height of the object at time t, where y0 is the
initial height (at time t=0), and v0 is its initial velocity.
51. Related Rates
Recall how related rates work. This is one of the big ideas that
makes calculus important:
If you know how z changes when y changes (dz/dy) and how y
changes when x changes (dy/dx), then you know how z changes
when x changes:
dz dz dy
=
dx dy dx
Remember the idea of implicit differentiation: The derivative of
f(y) with respect to x is f '(y)dy
dx
52. More on related rates
The idea is that "differentiating both
sides of an equation with respect to
x" [or any other variable] is a legal
(and useful!) operation.
This is best done by using examples...
53. Related Rates Greatest Hits
A light is at the top of a 16-ft pole. A boy 5 ft tall walks away from
the pole at a rate of 4 ft/sec. At what rate is the tip of his shadow
moving when he is 18 ft from the pole? At what rate is the length
A man on a dock is pulling in a boat by means of a rope attached
to the bow of the boat 1 ft above the water level and passing through
a simple pulley located on the dock 8 ft above water level. If he pulls
in the rope at a rate of 2 ft/sec, how fast is the boat approaching the
dock when the bow of the boat is 25 ft from a point on the water
directly below the pulley?
54. Greatest Hits...
A weather balloon is rising vertically at a rate of 2 ft/sec. An
observer is situated 100 yds from a point on the ground directly
below the balloon. At what rate is the distance between the balloon
and the observer changing when the altitude of the balloon is 500 ft?
The ends of a water trough 8 ft long are equilateral triangles whose
sides are 2 ft long. If water is being pumped into the trough at a rate
of 5 cu ft/min, find the rate at which the water level is rising when the
depth is 8 in.
Gas is escaping from a spherical balloon at a rate of 10 cu ft/hr. At
what rate is the radius chaing when the volume is 400 cu ft?
55. Check the WEB for
assignments and other
course information!
EMAIL [email protected]
in case of difficulty!
Next week: INTEGRALS!