Chapter 4.9,5,6 Flashcards
What is the antiderivative of m(x)=a?
M(x)=ax+C
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)
What is the antiderivative of a function x^-1?
ln|x|+C
What is the antiderivative of a function sinx?
-cosx+C
What is the antiderivative of a function cosx?
sinx+C
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)
What is the antiderivative of a function 1/(1+x^2)?
tan^-1(x)+C
What is the antiderivative of a function sec(x)tan(x)?
sec(x)+C
What is the equation for Δx when uniformly partitioned on a closed interval?
Δx=(b-a)/n
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)
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
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
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))
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
What is the antiderivative of f(x)=9x^e?
F(x)= (9x^(e+1)) / (e+1)
What is the antiderivative of f(x)=5e^x?
F(x)=5e^x+C
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
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)
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)
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
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)
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)
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.
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)
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)
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
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.
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
How do you find the average value of a function on an interval?
1/(b-a) times the Definite Integral
What is the Mean Value Theorem for Definite Integrals?
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
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)?
g’(x)=f(h(x))*h’(x)
