Calculus and Geometry Practice Flashcards

A set of practice flashcards covering derivative rules, antiderivatives, and geometric formulas for volume and surface area.

Derivative of $f(x) = x^3$

$$3x^2$$

Derivative of a constant function $f(x) = 5$

$$0$$

Derivative of $f(x) = \sqrt{x}$

$$\frac{1}{2\sqrt{x}}$$

Derivative of $7x^4 - 3x + 8$

$$28x^3 - 3$$

Derivative of $f(x) = \ln(x)$

$$\frac{1}{x}$$

Derivative of $e^x$

$$e^x$$

Derivative of $\sin(x)$

$$\cos(x)$$

Derivative of $\tan(x)$

$$\sec^2(x)$$

Quotient Rule for $y = \frac{x+1}{x-1}$

$$y' = \frac{(1)(x-1) - (x+1)(1)}{(x-1)^2}$$

Chain Rule Use Case

Differentiating a composition of functions, such as $\sin(x^2)$

Critical points

Occur when $f'(x) = 0$ or $f'(x)$ is undefined

Second derivative information

Indicates the concavity of a function

Inflection point

A point where $f''(x) = 0$ and the concavity of the graph changes

Implicit derivative of $x^2 + y^2 = 1$

$$y' = -\frac{x}{y}$$

Derivative of $\ln|x|$

$$\frac{1}{x}$$

Antiderivative of $x^3$ $\int x^3 \,dx$

$$\frac{x^4}{4} + C$$

Antiderivative of $\frac{1}{x}$ $\int \frac{1}{x} \,dx$

$$\ln|x| + C$$

Antiderivative of $\tan(x)$ $\int \tan(x) \,dx$

$\ln|\sec(x)| + C$ or $-\ln|\cos(x)| + C$

Volume of a cylinder

$$V = \pi r^2 h$$

Surface area of a sphere

$$SA = 4\pi r^2$$

Volume of a cone

$$V = \frac{1}{3} \pi r^2 h$$

Surface area of a cube

$$SA = 6a^2$$

Lateral surface area of a cone

$$\pi r l$$

Slant height

Used in a cone to calculate the lateral surface area

Ratio of cylinder volume to cone volume

$3:1$ when they share the same base and height

One-faced solid

A cone has only one flat face (the base)