Chapter 4.9,5,6

Question 1

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

Answer 1

M(x)=ax+C

Question 2

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

Answer 2

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

Question 3

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

Answer 3

ln|x|+C

Question 4

What is the antiderivative of a function sinx?

Answer 4

-cosx+C

Question 5

What is the antiderivative of a function cosx?

Answer 5

sinx+C

Question 6

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

Answer 6

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

Question 7

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

Answer 7

tan^-1(x)+C

Question 8

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

Answer 8

sec(x)+C

Question 9

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

Answer 9

Δx=(b-a)/n

Question 10

What is the definition for the definite interval of f from a to b? b
∫ f(x)dx =?
a

Answer 10

n
lim ∑ f(Xi*)Δx provided the limit exists
n->∞ i=1

(The limit of a Riemann Sum)

Question 11

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

Answer 11

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

Question 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?

Answer 12

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

Question 13

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

Answer 13

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

Question 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?

Answer 14

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

Question 15

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

Answer 15

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

Question 16

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

Answer 16

F(x)=5e^x+C

Question 17

What is Part 1 of the Fundamental Theory of Calculus?

Answer 17

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

Question 18

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

Answer 18

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)

Question 19

What is Part 2 of the Fundamental Theory of Calculus?

Answer 19

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

Question 20

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

Answer 20

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

Question 21

What is the Net Change Theorem?

Answer 21

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

Question 22

What is the Net Change Theorem?

Answer 22

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)

Question 23

What is the Substitution Technique of Integration?

Answer 23

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.

Question 24

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

Answer 24

=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)

Question 25

How do you find the area between two curves?

Answer 25

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)

Question 26

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

Answer 26

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

Question 27

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

Answer 27

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.

Question 28

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

Answer 28

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

Question 29

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

Answer 29

1/(b-a) times the Definite Integral

Question 30

What is the Mean Value Theorem for Definite Integrals?

Answer 30

For a continuous function on a closed interval [a,b], there is at least 1 c in [a,b] such that f(c)=1/b-a times the Definite Integral

Question 31

If g(x) is a Definite Integral f(t) with a constant for a lower base and a function h(x) for an upper base, what is g’(x)?

Answer 31

g’(x)=f(h(x))*h’(x)