Calculus Review Flashcards

These flashcards cover key concepts in calculus such as the Fundamental Theorems, the Mean Value Theorem, definitions of limits and derivatives, and concepts related to the area under curves.

What is the First Fundamental Theorem of Calculus (1st FTC)?

If f is continuous on [a, b], and F is the antiderivative of f on [a, b], then $\int_a^b f(x) \, dx = F(b) - F(a)$.

What does the Mean Value Theorem (MVT) state?

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

What is implied by the Intermediate Value Theorem (IVT)?

If f is continuous on [a, b], and k is between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = k.

How do you find the average value of a function f on the interval [a, b]?

The average value is given by $\frac{1}{b - a} \int_a^b f(x) \, dx$.

What is the definition of the derivative of a function f at a point a?

The derivative is given by $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$.

What does the Second Fundamental Theorem of Calculus (2nd FTC) state?

If f is continuous on [a, b], then the function G(x) defined by $G(x) = \int_a^x f(t) \, dt$ has a derivative G'(x) = f(x) for every point in [a, b].

What is the formula for the area between two curves f(x) and g(x) from a to b?

The area is given by $\int_a^b [f(x) - g(x)] \, dx$ where f(x) is the upper curve and g(x) is the lower curve.

State the formal definition of a limit.

A limit of f(x) as x approaches c exists if $\lim_{x \to c} f(x) = L$ where L is some finite number.

What are the types of discontinuities in functions?

There are three types: Point discontinuity (removable), Jump discontinuity, and Infinite discontinuity.

What is the relationship between the velocity and acceleration of an object?

If v(t) and a(t) have the same sign, the speed is increasing; if they have opposite signs, the speed is decreasing.