Intro to Calculus

Vocabulary and key formulas from the Intro to Calculus lecture notes covering differentiation rules, gradients, and curve behavior.

acceleration

The rate of change of velocity with respect to time.

average rate of change

The rate of change between 2 points on a function; the gradient of the line (secant) passing through those points.

chain rule

A method for differentiating composite functions using the formula $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$ or $h'(x) = f'(g(x))g'(x)$.

derivative function

The gradient function $y = f'(x)$ of a function $y = f(x)$ obtained through differentiation.

differentiability

A function is differentiable wherever its gradient is defined. A function is differentiable at point $x = a$ if its graph is continuous and smooth at that point.

differentiation

The process of finding the gradient function.

differentiation from first principles

The process of finding the gradient of a tangent to a curve by finding the gradient of the secant between 2 points and finding the limit as the secant becomes a tangent: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$.

displacement

The distance and direction of an object in relation to the origin.

gradient of a secant

The gradient (slope) of the line between 2 points on a function; measures the average rate of change between the 2 points.

gradient of a tangent

The gradient of a line that is a tangent to the curve at a point on a function; measures the instantaneous rate of change of the function at that point.

instantaneous rate of change

The rate of change at a particular point on a function; the gradient of the tangent at this point.

limit

The value that a function approaches as the independent variable approaches some value.

normal

A line that is perpendicular to the tangent at a given point on a curve ($m_1 m_2 = -1$).

product rule

A method for differentiating the product of 2 functions: if $h(x) = f(x)g(x)$, then $h'(x) = f(x)g'(x) + f'(x)g(x)$, or $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$.

quotient rule

A method for differentiating the quotient of 2 functions: if $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$, or $\frac{d}{dx}(\frac{u}{v}) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.

secant

A straight line passing through 2 points on the graph of a function.

stationary point

A point on the graph of $y = f(x)$ where the tangent is horizontal and its gradient $f'(x) = 0$. It could be a maximum point, minimum point, or a horizontal point of inflection.

tangent

A straight line that just touches a curve at one point. The curve has the same gradient or direction as the tangent at that point.

turning point

A maximum or minimum point on a curve, where the curve turns around.

velocity

The rate of change of displacement of an object with respect to time; involves speed and direction.

angle of inclination

The relationship between the angle of inclination of a line, $\theta$, with the positive $x$-axis and the gradient $m$, established as $\tan(\theta) = m$.

difference quotient

The formula $\frac{f(x+h)-f(x)}{h}$ which represents the average rate of change of $f(x)$ or the gradient of a chord or secant.

derivative notation

Symbols used to represent the derivative, including $\frac{dy}{dx}$, $f'(x)$, and $y'$.

power rule

The differentiation formula $\frac{d}{dx}(x^n) = nx^{n-1}$ for all real values of $n$.

stationary point condition

The condition where the gradient function $f'(x) = 0$, meaning the gradient function is on the $x$-axis.

increasing function condition

A function is increasing when the gradient $f'(x) > 0$, meaning the gradient function is above the $x$-axis.

decreasing function condition

A function is decreasing when the gradient $f'(x) < 0$, meaning the gradient function is below the $x$-axis.

derivative of a constant

The rule stating that $\frac{d}{dx}(k) = 0$, where $k$ is a constant.

derivative of kx

The rule stating that $\frac{d}{dx}(kx) = k$.

derivative of a constant multiple

The rule stating that $\frac{d}{dx}(kf(x)) = kf'(x)$.