Math 1190 Exam 2 Review: Derivative Rules, Business Applications, and Curve Sketching

Vocabulary practice flashcards covering derivative rules, business applications, curve sketching, concavity, and optimization for Math 1190 Exam 2.

Elasticity of Demand, $E(p)$

A measure of the sensitivity of the quantity demanded to changes in price, given by the formula $E(p) = -\frac{p \times D'(p)}{D(p)}$.

Inelastic Demand

A condition occurring when $0 < E(p) < 1$, meaning demand is relatively insensitive to price changes; to increase total revenue in this state, price should be increased.

Elastic Demand

A condition occurring when $E(p) > 1$, meaning demand is sensitive to price changes; decreasing price will increase total revenue.

Relative Rate of Change

The ratio of the instantaneous rate of change of a function to the value of the function itself, represented as $\frac{f'(x)}{f(x)}$.

Marginal Revenue

The instantaneous rate of change of total revenue with respect to the quantity sold, calculated as the first derivative of the revenue function, $R'(q)$.

Product Rule

A differentiation rule stating that the derivative of a product of two differentiable functions is given by $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$.

Quotient Rule

A differentiation rule stating that the derivative of a quotient of two differentiable functions is given by $\frac{d}{dx}\begin{pmatrix}\frac{f(x)}{g(x)}\end{pmatrix} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.

Chain Rule

A differentiation rule for composite functions $h(x) = f(g(x))$ stating that the derivative is the outer derivative evaluated at the inner function times the inner derivative: $h'(x) = f'(g(x)) \times g'(x)$.

Critical Value

A value $c$ in the domain of a function $f(x)$ where either the first derivative equals zero ($f'(c) = 0$) or the first derivative is undefined.

Increasing Function Condition

A relationship establishing that a function $f(x)$ is increasing on an open interval wherever its derivative is strictly positive ($f'(x) > 0$).

Decreasing Function Condition

A relationship establishing that a function $f(x)$ is decreasing on an open interval wherever its derivative is strictly negative ($f'(x) < 0$).

Horizontal Tangent Line

A tangent line with a slope of zero, occurring at any point on a differentiable curve where the first derivative equals zero ($f'(x) = 0$).

First Derivative Test

A method to classify local extrema by testing sign changes of $f'(x)$ around a critical value: changing from positive to negative indicates a local maximum, while changing from negative to positive indicates a local minimum.

Inflection Point

A point on a graph of a continuous function where the concavity changes from concave up to concave down, or from concave down to concave up.

Concavity Test

A test determining the bending of a graph: the graph is concave upward where the second derivative satisfies $f''(x) > 0$, and concave downward where $f''(x) < 0$.

Second Derivative Test

A test used to classify critical values where $f'(c) = 0$: if $f''(c) > 0$, the point is a local minimum; if $f''(c) < 0$, the point is a local maximum; if $f''(c) = 0$, the test is inconclusive.

Extreme Value Theorem Candidates

The set of points on a closed interval $[a, b]$ that could yield absolute extrema for a continuous function, consisting solely of the critical values within $(a, b)$ and the endpoints $a$ and $b$.