# Calculus 2 Exam 1

## 14 terms

### Definition 6.1 - Primitives or Antiderivatives

Let f be a function defined on an interval I. We say that a function G is a primitive of f if
1. G'(x)=f(x) for all x in the interior of I, and
2. G is continuous on I.
A synonym for primitive is antiderivative.

### Theorem 6.18 - Fundamental Theorem of Calculus

If f is a continuous function on a close interval [a,b], then
1. f has a primitive on [a,b]. In fact the function H(x)= the integral from a to x f(t)dt is such a primitive.
2. d/dx integral from a to x f(t)dt=f(x).
3. If G is any primitive of f on [a,b], then the integral from a to b f(x)dx=G(b)-G(a).

### Theorem 6.19 - Integration is a Linear Operation

Definite and Indefinite integration are linear operations. More precisely, if f and g are integrable functions with the same domains, then
1. the integral from a to b f(x)+g(x)dx= the integral from a to b f(x)dx + the integral from a to b g(x)dx
2. the integral from a to b cf(x)dx=c the integral from a to b f(x)dx if c is any constant, and
1. the integral f(x)+g(x)dx=the integral f(x)dx + the integral g(x)dx
2. the integral cf(x)dx=c the integral f(x)dx if c is any constant.

### Theorem 6.23 - Monotonicity Theorem for Integrals

Let f and g be integrable functions on the closed interval [a,b]. Then,
1. If f is > or = 0 on [a,b], then the integral from a to b f(x)dx is > or = 0.
2. If f is > or = g on [a,b], then the integral from a to b f(x)dx is > or = the integral from a to b g(x)dx. Likewise, if f is < or = g, then the integral from a to b f(x)dx is < or = the integral from a to b g(x)dx.
3. If m is < or = f(x) which is < or = M on [a,b], then m(b-a) is < or = the integral from a to b f(x)dx which is < or = M(b-a).
4. the absolute value of the integral from a to b f(x)dx is < or = the integral from a to b of the absolute value of f(x) dx.
5. If the absolute value of f(x) is < or = M, then the absolute value of the integral from a to b f(x)dx is < or = M(b-a).

### Theorem 6.27 - Riemann Sums and Integrals

If f is continuous on the close interval [a,b], then the integral from a to b f(x)dx = the lim from n to infinity of the Riemann sum where j=1 to n f(x subscript j) deltax. In words, the definite integral is the limit of the Riemann sums.

### Definition 6.28 - Average Value of a Function

If f is integrable on [a,b], then we define the average value of f on that interval to b avg(f)= 1/(b-a) times the integral from a to b f(x)dx.

### Theorem 6.33 - Integrability Theorem

1. Boundedness is a necessary condition for a function to be Riemann integrable. Hence if f is unbounded on [a,b], then f is not Riemann integrable there and the integral fro ma to b f(x)dx does not exist.
2. If f is bounded on the closed interval [a,b] and f has only finitely many discontinuities there, then f is Riemann integrable, and therefore the integral fro ma to b f(x)dx exists.
3. If f is bounded on the closed interval [a,b] and f is either increasing, or f is decreasing on that interval, then f is Riemann integrable, and therefore the integral from a to b f(x)dx exists.
This result encompasses continuous functions: every continuous function on a closed interval [a,b] is Riemann integrable.

### Definition 7.1 - Inverse Functions

Let f : D -> R be a real valued function. Then a function g : E -> R is called an inverse to the function f, if and only if f(g(y))=y and g(f(y))=x for all x which is an element of the domain(f) and all y which is an element of domain(g). This establishes the idea of the inverse g of a function f under the operation of function composition. We simply call g the inverse of f and we denote it by the symbol f to the -1.

### Definition 7.4 - One-to-One Function

A function f is said to b a one-to-one function if the implication x1 does not equal x2 => f(x1) does not equal f(x2) is true for all x1 and x2 in the domain of f. An equivalent condition for f to be one-to-one is f(x1)=f(x2)=>x1=x2 for all x1 and x2 in the domain of f.

### Definition 7.14 - The ln Function

For all x which is an element of 0 to infinity define ln(x)= the integral from 1 to x 1/t dt. This defines a function ln with domain 0 to infinity.

exp=ln to the -1

### Definition 7.21 - General Exponential Functions and Irrational Powers

Let a be a positive real number. For any real number x, define the power a to the x by the following rule : a to the x=exp(xln(a)). The function f(x)= a to the x with domain of negative infinity to infinity is called a general exponential function with base a.

### Theorem 7.28 - Derivatives and Integrals Involving Logarithms

Let a be a positive real number with a does not equal 1. d/dx (log of a(x))=1/ln(a) times 1/x where x is any number in the domain 0 to inifinity of the logarithm. Also, d/dx(a to the x)=ln(a) times a to the x. the integral a to x dx= 1/ln(a) times a to the x + c.

### Theorem 7.30 - Power Rule for Integrals

Let n be any real number, if n does not equal -1 then the integral (x to the n)dx= x to the n+1/n+1 + C. if n = -1 then the integral (x-1)dx = the integral 1/x dx= lnlxl + c.