Reimann Sums

1. From Power Rule to Riemann
Sums
A Journey through Limits, Derivatives, and
Integrals
Group 1
Xiaoshan Yu
Spandana Govindgari

2. Limits
Definition
We say that the limit of f(x) is L as x approaches a and write
this as
lim f(x) = L
x --> a
provided we can make f(x) as close to L as we want for all x
sufficiently close to a, from both sides, without actually letting x
be a.

3. Finding Limits
Graphically
-look at where graph is going
Analytically
1. Always substitute first
2. If the answer is a real number. that is the limit
3. If you get 0/0, simplify by factoring, multiply by conjugate
4. If you get a real number/0 - Vertical asymptote - Plug in a number a
little larger or smaller and check signs
One-Sided Limits
-as a function approaches a vertical asymptote, it either decreases or
increases without bound (+ or - infinity)
Two-Sided Limits
-if the two one-sided limits approaching the same point are equal, then
they are the two-sided limit

4. Finding Limits
Limits to Infinity
-find the limit to infinity
Numerically
-divide each term by the largest power term, then plug in to find
the limit

5. Finding Limits
L'Hopital's Rule
Indeterminate form
0/0, ∞/∞, ∞-∞, 0*∞, 0^0, 1^∞, ∞^0
Determinate form
∞+∞, -∞-∞, 0^∞
If a limit is indeterminate and in the form, 0/0 or ∞/∞, then the
limit is equal to the limit of the derivative of both the numerator
over the derivative of the denominator.

6. Analyzing Functions
Asymptotes
Vertical: zero of denominator/point where function is undefined
Horizontal:
-highest power in denominator is greater than that in
the numerator: y=0
-highest powers in denominator and numerator
equal: y=quotient of coefficients of terms with
highest powers
Slant:
-highest power in numerator is greater: divide numerator
by denominator with long division
Continuity
1. f(c) is defined
2. lim f(x) exists
x-->c
3. lin f(x) = f(c)
x-->c
Continuous everywhere if continuous at each point in domain.

7. Analyzing Functions
Differentiability
-not differentiable at:
●vertical asymptotes
●sharp turn
●cusp
●vertical tangent line
●discontinuities
If a function is differentiable at x=c, then the function is
continuous.

8. Analyzing Function
Tangent Line
-find values of dy/dx and x and y
-find equation of straight line using these values
Normal Line
-line perpendicular to tangent line
-slope is the negative reciprocal of tangent line
Horizontal Tangent Line
-where dy/dx=0
Vertical Tangent Line
-where dy/dx=undefined

9. Derivatives
Definition
-the instantaneous change of one quantity relative to another;
df(x)/dx
Product Rule
Quotient Rule
Chain Rule F'(x) = f '(g(x)) g '(x)
Power Rule d
dx xn = nxn−1
Symbolic Differentiation
-treat symbols as constants

10. Derivatives for Special Functions
Rational Functions
-use quotient rule
Polynomials
-use power rule, chain rule,
etc.
Trigonometric

11. Derivatives for Special Functions
Inverse Trigonometry
Exponential
Natural Logarithm

12. Derivatives for Special functions
Inverse
Method 1:
-switch x and y
-use implicit differentiation to solve dy/dx
-plug in values for inverse function to solve
Method 2:
-plug the x value of inverse function into the original function as
y to find x value for the original function
-find first derivative of original function
-plug this value of x into the first derivative
-the derivative of the inverse will be the reciprocal of the value
obtained

13. Analyzing Functions Using Derivatives
Increase, Decreasing
-find the first derivative and critical values by setting first
derivative to zero
-make a sign chart and find if each interval is positive or
negative
-positive = increasing
-negative = decreasing

14. Analyzing Functions Using Derivatives
Concavity
-find the second derivative and second derivative zeroes by
setting second derivative to zero
-make a sign chart
-positive = concave up
-negative = concave down
Inflection Points
-points at which the concavity change

15. Applications
Rate of Change or Slope- the change in y over x
Related Rates: When one quantity increases the other
decreases simultaneously or increases simultaneously. The
rates of change of two quanities are related by their algebraic
relationship
Velocity - distance over time
Instantaneous Rate of Change vs. Average Rate of change
Instantaneous - find derivative of the function at that specific
point
Average - The slope is found over an interval and hence is not
a closer approximation to the instantaneous rate of change

16. Antidifferentiation = Integration
Definition
A function F(x) is an antiderivative of a function f (x) if F'(x) =
f(x) for all x in the domain of f.
The process of finding an antiderivative is antidifferentiation.
Types of Integrals
Definite - has to be integrated over an interval [a, b]
[Second Fundamental Theorem of Calculus]
Indefinite - has no limits of integration (need a + C)
= F(x) + C

17. Techniques and Formulae
u- subsitution
works if x has a coefficient or if x is raised to a power
Power rule
add 1 to the power and divide by the new power
Logarithmic
Exponential - There is a man who got on the bus and he was
threatening people saying, "I will derive you and I will integrate
you" then everyone got off the bus except for one person. Who
is he?

18. Properties of Integrals
Addition Reversing limits of integration
Subtraction Integrating over the
same limits

19. Riemann Sums
Definite Integral is a Limit of Riemann Sum
Riemann Sum is an approximation of an integral
Use rectangles to approximate the area under a curve
Widths are the same
As the number of intervals used in a Riemann sum approaches infinity,
the approximation approaches the value of the definite integral.
Left side
The left vertex of the rectangle is used as length
Right side
The right vertex of the rectangle is used as length
Midpoint
Use the y value for the x value in the middle of the interval of the
rectangle
Trapezoidal Rule
Area = w/2[2f(x1) + f(x2) + .........+ 2f (xn)]
Use trapezoids instead of rectangles
Concave down: Underestimates
Concave up: Overestimates

20. Mean Value Theorem
Rate of Change of Quantity of Interval (Average)
The average value of a function f(x) is taken over an interval [a,
b]
Average =
This also gives the height of the mean value rectangle.
C, where f(c) equals the average, is the mean value.

21. Improper Integrals
Infinite Limits of Integration
1.if f is continuous on [a,∞) then
2.if f is continuous on (-∞,b] then
3.if f is continuous on (-∞,∞) then
Infinite Discontinuities
1.if f is continuous on [a,b) and has infinite discontinuity at b
2. if f is continuous on (a,b] and has infinite discontinuity at a
3. if f is continuous on [a,b] except at c at which f has
infinite discontinuity both integrals
on right must converge for the left to converge