![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](https://image.slidesharecdn.com/calcbcpresentationprojectthingie-190313220524/85/Calculus-BC-16-320.jpg)
![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](https://image.slidesharecdn.com/calcbcpresentationprojectthingie-190313220524/85/Calculus-BC-19-320.jpg)
![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.](https://image.slidesharecdn.com/calcbcpresentationprojectthingie-190313220524/85/Calculus-BC-20-320.jpg)
![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](https://image.slidesharecdn.com/calcbcpresentationprojectthingie-190313220524/85/Calculus-BC-21-320.jpg)
Calculus BC
- 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
Editor's Notes
- #20: Hi