Calculus Review Notes

A set of vocabulary flashcards covering key derivatives, theorems, and calculus concepts.

Basic Derivative of a Power Function

The derivative of $x^n$ is $n x^{n-1}$.

Basic Derivative of Sine

The derivative of $\sin(x)$ is $\cos(x)$.

Basic Derivative of Cosine

The derivative of $\cos(x)$ is $-\sin(x)$.

Basic Derivative of Tangent

The derivative of $\tan(x)$ is $\sec^2(x)$.

Basic Derivative of Cotangent

The derivative of $\cot(x)$ is $-\csc^2(x)$.

Basic Derivative of Secant

The derivative of $\sec(x)$ is $\sec(x)\tan(x)$.

Basic Derivative of Cosecant

The derivative of $\csc(x)$ is $-\csc(x)\cot(x)$.

Basic Derivative of Natural Logarithm

The derivative of $\ln(u)$ is $\frac{1}{u}$.

Basic Derivative of Exponential Function

The derivative of $e^u$ is $e^u$.

Chain Rule

If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = f'(u)\frac{du}{dx}$.

Product Rule

If $y = u v$, then $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$.

Quotient Rule

If $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$.

Intermediate Value Theorem

If $f(x)$ is continuous on $[a,b]$, and $y$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a,b)$ such that $f(c) = y$.

Mean Value Theorem

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

Rolle's Theorem

If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c$ in $(a,b)$ such that $f'(c) = 0$.

Extreme Value Theorem

If $f(x)$ is continuous on $[a,b]$, then $f(x)$ has an absolute maximum and minimum on that interval.

Average Rate of Change (ARoC)

$m_{ARoC} = \frac{f(b) - f(a)}{b - a}$.

Instantaneous Rate of Change (IRoC)

$m_{IRoC} = f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

Tangent Line Equation

The equation is $y - y_1 = m(x - x_1)$ where $m$ is the slope.

First Derivative Test

If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.

Second Derivative Test

If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Fundamental Theorem of Calculus

$\int_a^b f(x) \, dx = F(b) - F(a)$ where $F' = f$.

Distance, Velocity, Acceleration Relationship

The derivative of the position function $x(t)$ is the velocity function $v(t)$.

Displacement Formula

The displacement from $t_0$ to $t_f$ is $\int_{t_0}^{t_f} v(t) \, dt$.

Average Velocity Formula

Average velocity is $\frac{\Delta x}{\Delta t}$.