MAT 131 Calculus I Practice Exam 2 Vocabulary Review

Constant of Integration ($+C$)

An arbitrary constant added to the antiderivative in an indefinite integral to represent the entire family of antiderivatives, which is omitted in definite integrals.

Indefinite Integral

The collection of all antiderivatives of a function $f(x)$, written as $\int f(x)\,dx = F(x) + C$, where $F'(x) = f(x)$.

Upper Sum

A Riemann sum approximation of the area under a curve or total accumulated amount obtained by partitioning an interval into subintervals and evaluating the function at its maximum value on each subinterval.

Work (Calculus)

The accumulation of a continuous variable force $F(x)$ exerted over a straight-line displacement from $x = a$ to $x = b$, defined by the definite integral $W = \int_a^b F(x)\,dx$ with units in joules.

Interval Additivity Property of Integrals

The integral property stating that for an integrable function $f$ and any points $a$, $b$, and $c$, $\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$, which implies $\int_6^4 F(x)\,dx = \int_1^4 F(x)\,dx - \int_1^6 F(x)\,dx$.

Average Rate of Change

The mean rate at which a function $P(t)$ changes over a closed interval $[a, b]$, given by the expression $\frac{1}{b-a}\int_a^b P'(t)\,dt = \frac{P(b) - P(a)}{b-a}$.

Harmonic Oscillator Differential Equation

A second-order differential equation of the form $y'' = -ky$ (where $k > 0$), modeling a mass oscillating about equilibrium where acceleration is proportional to displacement and directed toward equilibrium.

Slope Field of $\frac{dy}{dx} = -\frac{x}{y}$

A direction field where line segments are horizontal along the $y$-axis ($x = 0$), undefined along the $x$-axis ($y = 0$), have negative slopes in Quadrants I and III, and have positive slopes in Quadrants II and IV.

Logarithmic Differentiation

A method of differentiation used for functions where both the base and exponent contain variables, such as $y = x^{\sqrt{x}}$, by taking the natural logarithm of both sides before differentiating implicitly.

Linear Density

The mass per unit length of a one-dimensional object, denoted $r(x)$ in $\text{kg/m}$, whose definite integral $\int_a^b r(x)\,dx$ yields the total mass of the object over the interval $[a, b]$.

Net Change Theorem

A formulation of the Fundamental Theorem of Calculus stating that the definite integral of a rate of change gives the net change of the original quantity: $\int_a^b r'(x)\,dx = r(b) - r(a)$.

Displacement

The net change in position of an object over a time interval $[a, b]$, calculated by the definite integral of its velocity function: $\int_a^b v(t)\,dt$.

Total Distance Traveled

The total length of the path traveled by an object across time interval $[a, b]$, calculated by the definite integral of speed (the absolute value of velocity): $\int_a^b |v(t)|\,dt$.

Average Value of a Function

The mean height of a continuous function $f(x)$ over an interval $[a, b]$, given by the formula $\frac{1}{b-a}\int_a^b f(x)\,dx$.