Improper Integral Concepts: AP Calculus BC

lim x→±∞ (1/Xⁿ) = (when n is positive)

0

What makes and integral improper?

1) If one or both bounds (a and b) is ±∞. 2) If a or b is an infinite discontinuity of f 3) There is a value (c) between a and b such that c is a infinite discontinuity of f.

Limits at infinity of rational functions (Where the degree of the numerator is N and the degree of the denominator is D)

1) When N<D, lim = 0 2) When N=D, lim = quotient of leading coefficients 3) When N>D, lim = ±∞ (depending on the simplified rational expression)

does 1/e^x converge or diverge?

Converges

Limit Comparison Test (LCT)

If f(x) and g(x) are non-negative and they grow at the same rate, then either ∫from a→∞ f(x) and ∫from a→∞ g(x) BOTH converge or BOTH diverge.

What must you do when the a value c between bounds of integration a and b causes a discontinuity in f?

Split the integral into two separate integrals.

For LCT, how do you determine if the growth rates of g(x) and f(x) are equal?

Steps: 1) Divide the two functions f(x)/g(x). 2) Take the limit at ∞ of the quotient. 3) If the limit exists, but is not 0, f(x) and g(x) grow at the same rate.

What makes an improper integral convergent?

The limit exists and in finite.

What makes and improper integral divergent?

The limit is ±∞

DCT: If f and g are continuous on [a, ∞) and 0 ≤ f(x) ≤ g(x) for x≥a (f is less than or equal to g) and g(x) converges, what does that mean for f(x)

f(x) also converges

DCT: If f and g are continuous on [a, ∞) and 0 ≤ f(x) ≤ g(x) for x≥a (g is greater than or equal to f) and f(x) diverges, what doe that mean for g(x)

g(x) also diverges

In the case of ∫(from 0→1) of (a/xⁿ) dx What happens when n<1, n=1, and n>1?

n<1→ Convergent n=1→ Divergent n>1→ Divergent

In the case of ∫(from 1→∞) of (a/xⁿ) dx What happens when n<1, n=1, and n>1?

n<1→Divergent n=1→Divergent n>1→Convergent