Calculus Lecture Notes Review

Flashcards covering key concepts from lectures on limits, derivative rules, and fundamental theorems in calculus.

How do you evaluate the limit of (x³ - 27) / (x - 3) as x approaches 3?

Factor the numerator into (x - 3)(x² + 3x + 9), cancel the (x - 3) terms, and then plug in x = 3 to get 27.

What technique should you use when complex fractions appear in a limit expression?

Multiply the numerator and the denominator by the common denominator.

What technique should you use when dealing with square roots in a limit problem?

Multiply the numerator and the denominator by the conjugate of the expression containing the square root.

What is the limit of sin(x) / x as x approaches 0?

1

What is the limit of (1 - cos(x)) / x as x approaches 0?

0

What are the first steps to try when evaluating a limit?

First, attempt to plug in the value if the function is continuous. If that doesn't work, try algebraic manipulation, or consider special cases.

State the Squeeze Theorem.

If g
,(x)
≤
f(x)
≤
g₂(x) for all x in an open interval containing c (except possibly at c), and lim g
,(x) = L = lim g₂(x) as x approaches c, then lim f(x) = L as x approaches c.

Using the Squeeze Theorem, what is the limit of x³ cos(1/x) as x approaches 0?

The limit is 0, because -x³ ≤ x³ cos(1/x) ≤ x³, and as x approaches 0, both lim(-x³) = 0 and lim(x³) = 0.

State the Intermediate Value Theorem (IVT).

If f(x) is continuous on a closed interval [a,b] and p is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = p.

State the Power Rule for derivatives.

The derivative of xⁿ is nxⁿ⁻¹.

State the Product Rule for derivatives for two functions f(x) and g(x).

The derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x).

State the Quotient Rule for derivatives for f(x)/g(x).

The derivative of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x)) / (g(x))².

State the Chain Rule for derivatives for a composite function F(g(x)).

The derivative of F(g(x)) is F'(g(x)) * g'(x).

What is the alternative definition of a derivative in terms of a limit (f'(a) form)?

f'(a) = lim (f(x) - f(a)) / (x - a) as x approaches a.

What are the three fundamental Pythagorean trigonometric identities?

sin²x + cos²x = 1; 1 + cot²x = csc²x; tan²x + 1 = sec²x.