Final Review

Front

What is the definition of continuity at a point?

Back

A function f(x) is continuous at x=a if: 1) f(a) is defined, 2) \( \lim_{x \to a} f(x) \) exists, and 3) \( \lim_{x \to a} f(x) = f(a) \).

Front

What does the limit of a function represent?

Back

The limit of a function as x approaches a certain value represents the value that the function approaches as x gets arbitrarily close to that value.

Front

What is the Power Rule in differentiation?

Back

The Power Rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

Front

What is the Intermediate Value Theorem (IVT)?

Back

The IVT states that if a function is continuous on the interval [a, b], and f(a) and f(b) have different signs, then there exists at least one c in (a, b) such that f(c) = 0.

Front

What is a limit that does not exist (DNE)?

Back

A limit is said to not exist if the function approaches different values from the left and right sides, or if it approaches infinity.

Front

What is the definition of a zero of a function?

Back

A zero of a function f(x) is a value x=a such that f(a) = 0.

Front

What is the Quotient Rule in differentiation?

Back

The Quotient Rule states that if \( f(x) = \frac{g(x)}{h(x)} \), then \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \).

Front

What is the Chain Rule in differentiation?

Back

The Chain Rule states that if a function is composed of two functions, say f(g(x)), then \( f'(g(x)) = f'(g(x)) \cdot g'(x) \).

Front

What is the Product Rule in differentiation?

Back

The Product Rule states that if \( f(x) = g(x) \cdot h(x) \), then \( f'(x) = g'(x)h(x) + g(x)h'(x) \).

Front

What does it mean for a function to be continuous on an interval?

Back

A function is continuous on an interval if it is continuous at every point in that interval.