Flashcards in Chapter 4.9,5,6 Deck (31)

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1

## What is the antiderivative of m(x)=a?

### M(x)=ax+C

2

## What is the antiderivative of a function x^n? (when n is not -1)

### (1/n+1)x^n+1+C (Ex: j(x)=x^4, J(x)=1/5x^5+C)

3

## What is the antiderivative of a function x^-1?

### ln|x|+C

4

## What is the antiderivative of a function sinx?

### -cosx+C

5

## What is the antiderivative of a function cosx?

### sinx+C

6

## What is the antiderivative of a function b^x? (when b is a constant)

### b^x/lnx+C (The derivative of b^x is b^x*lnx)

7

## What is the antiderivative of a function 1/(1+x^2)?

### tan^-1(x)+C

8

## What is the antiderivative of a function sec(x)tan(x)?

### sec(x)+C

9

## What is the equation for Δx when uniformly partitioned on a closed interval?

### Δx=(b-a)/n

10

##
What is the definition for the definite interval of f from a to b? b

∫ f(x)dx =?

a

###
n

lim ∑ f(Xi*)Δx provided the limit exists

n->∞ i=1

(The limit of a Riemann Sum)

11

## What is the property of a Definite Integral where the upper and lower bounds are both a?

### A definite Interval where the upper and lower bounds are both the same number has a width of 0 and thus equals 0

12

## What is the relation between a Definite Integral where the upper bound is b and the lower bound is a, and a Definite Interval where the upper bound is a and the lower bound is b?

### If the upper and lower bounds are flipped, the definite integral is equal to negative the definite integral where the upper and lower bounds aren't flipped

13

## What is the property of a Definite Integral (f(x)+g(x))?

### That Definite Integral is equal to the definite integral(f(x)) +the definite integral(g(x))

14

## What is the property involving a Definite Integral with lower bound a and upper bound c + a Definite Integral with lower bound c and upper bound b?

### That equals a definite integral with lower bound a and upper bound b

15

## What is the antiderivative of f(x)=9x^e?

### F(x)= (9x^(e+1)) / (e+1)

16

## What is the antiderivative of f(x)=5e^x?

### F(x)=5e^x+C

17

## What is Part 1 of the Fundamental Theory of Calculus?

###
For g(x)= a∫^x f(t)dt, we find g'(x)=f(x)

THIS IMPLIES THAT INTEGRATION AND DIFFERENTIATION ARE INVERSE OPERATORS (with upper bound x and lower bound a) AND THAT EVERY CONTINUOUS F(X) ON A CLOSED INTERVAL HAS AN ANTIDERIVATIVE

18

## What do you use Part 1 of the Fundamental Theory of Calculus for?

### Can find an f'(x) of f(x)=a∫^x f(t)dt by manipulating the function using the properties of definite intervals (and knowledge of Part 1 FTC)

19

## What is Part 2 of the Fundamental Theory of Calculus?

### a∫^b f(x)dx=F(b)-F(a) where F(x) is any AD of f(x)

20

## What is the indefinite integral of f and how is it used?

### It is ∫f(x)dx (no bounds) and is used to ask for the antiderivative

21

## What is the Net Change Theorem?

### The Definite Integral of a Rate of Change is the Net Change ie: a∫^b f'(x)dx=f(b)-f(a)

22

## What is the Net Change Theorem?

### The Definite Integral of a Rate of Change is the Net Change ie: a∫^b f'(x)dx=f(b)-f(a) (Ex with bees and velocity)

23

## What is the Substitution Technique of Integration?

### With a given Integral, can substitute a function (w) and the derivative of that function (dw) for parts of the original integral. If a definite Integral, then can evaluate area using the 2nd part of the Fundamental Theory of Calculus.

24

## How do you find the area between f and x-axis? (In contrast to finding the "net area")

### =a∫^b |f(x)| dx=a∫^c |f(x)|dx +c∫^b |f(x)|dx= |a∫^c f(x)dx|+|c∫^b f(x)dx| (Find the absolute value of both parts of the Integral, with each part being separated by the x-intercept point c)

25

## How do you find the area between two curves?

### a∫^b |f(x)-g(x)| dx or |a∫^c f(x)-g(x)dx| + c∫^b |f(x)-g(x)|dx (If the functions intercept use the point c to separate them at the intercept point)

26

## What is the disk method and when do you use it?

### Finding the volume of a shape by finding the Definite Integral of (pi(radius)^2) Use it when you have a one to one function rotated by a line (touching the line of rotation) The slice being integrated is perpendicular to the axis of rotation

27

## What is the washer method and when do you use it?

### Finding the volume of a shape by finding the Definite Integral of (pi(outer) radius)^2-(pi(inner radius)^2.Use it when the shape is not touching the line of rotation. The slice being integrated is perpendicular to the axis of rotation.

28

## What is the shell method and when do you use it?

### Finding the volume of a shape by finding the Definite integral of (2pi(x)(f(x)) with x being the radius and f(x) the height(in relation to the slice) The slice being integrated is parallel to the axis of rotation

29

## How do you find the average value of a function on an interval?

### 1/(b-a) times the Definite Integral

30