Derivatives and Calculus Concepts

These flashcards cover essential derivatives and calculus concepts essential for understanding calculus at the AB/BC level.

What is the definition of the derivative of a function f(x)?

The derivative is defined as $rac{df(x)}{dx} = ext{lim}_{h o 0} \frac{f(x + h) - f(x)}{h}$.

What theorem guarantees there is at least one number c in (a,b) such that f(c) = 0 if f(a) = f(b)?

Rolle's Theorem.

What does the Intermediate Value Theorem state about continuous functions?

For any value between f(a) and f(b), there exists at least one point in the interval [a, b] where the function takes that value.

What is the formula for the Product Rule in differentiation?

$\frac{d(uv)}{dx} = u'v + uv'$.

What conditions must a function satisfy to be continuous at x = a?

1. f(a) is defined, 2. $\text{lim}<em>{x \to a} f(x) = f(a)$, 3. $\text{lim}</em>{x \to a} f(x)$ exists.

What is the formula for the Area Under the Curve using the Fundamental Theorem of Calculus?

$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$, where F is an antiderivative of f.

What is the quotient rule formula for differentiation?

$\frac{d(\frac{u}{v})}{dx} = \frac{v \cdot u' - u \cdot v'}{v^2}$.

What is the Chain Rule used for in calculus?

It is used to differentiate composite functions, expressed as $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$.

What is the formula for the Mean Value Theorem?

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

How is average velocity defined in terms of displacement?

Average velocity = $\frac{\Delta s}{\Delta t}$, where $\Delta s$ is the change in position and $\Delta t$ is the change in time.

What is the definition of critical points of a function f(x)?

Critical points are where f'(x) = 0 or f'(x) is undefined.

What does the Extreme Value Theorem state about a continuous function on a closed interval [a, b]?

A continuous function on [a, b] must have both an absolute maximum and an absolute minimum.

What is the formula for the area of a solid of revolution using the Disk Method?

$V = \pi \int [R(x)]^2 \, dx$.

What does l'Hôpital's Rule apply to?

l'Hôpital's Rule is applied when evaluating limits of the form 0/0 or ∞/∞.

What is the Taylor Series expansion of a function centered at x=a?

$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + … + \frac{f^{(n)}(a)}{n!}(x - a)^n$.