Calculus AB Exam 2024 S1 part 2

Front

What is the definition of a limit in calculus?

Back

A limit is a value that a function approaches as the input approaches some value. It is denoted as \( \lim_{x \to a} f(x) \).

Front

What does it mean for a function to be continuous at a point?

Back

A function is continuous at a point \( a \) 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

How do you find the derivative of a function?

Back

The derivative of a function \( f(x) \) at a point is the limit of the average rate of change of the function as the interval approaches zero: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

Front

What is the equation of a normal line to a curve at a given point?

Back

The equation of the normal line at point \( (a, f(a)) \) is given by: \( y - f(a) = -\frac{1}{f'(a)}(x - a) \).

Front

What are critical points of a function?

Back

Critical points occur where the derivative is zero or undefined. They are potential locations for local maxima, minima, or points of inflection.

Front

How do you determine concavity of a function?

Back

Concavity is determined by the second derivative: if \( f''(x) > 0 \), the function is concave up; if \( f''(x) < 0 \), it is concave down.

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus states that if \( f \) is continuous on \([a, b]\), then \( \int_a^b f(x)dx = F(b) - F(a) \), where \( F \) is an antiderivative of \( f \).

Front

What is an antiderivative?

Back

An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \).

Front

How do you find the area under a curve?

Back

The area under a curve from \( a \) to \( b \) is found using the definite integral: \( A = \int_a^b f(x)dx \).

Front

What is the power rule for differentiation?

Back

The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).