Calculus I Exam Concepts and Terminology

Flashcards testing vocabulary, fundamental calculus theorems, derivatives, and curve analysis terms based on the provided exam booklet.

Absolute Maximum Value

The greatest value achieved by a function on a specified interval $[a,b]$. For $f(x) = x^3 - 3x + 5$ on $[0,3]$, the absolute maximum value is $23$ occurring at $x = 3$.

Derivative of Arctangent

The rule for differentiating an inverse tangent function, given by $\frac{d}{dx}\big(\tan^{-1}(u)\big) = \frac{1}{1+u^2} \frac{du}{dx}$. For $y = \tan^{-1}(x^3)$, the derivative is $\frac{3x^2}{1+x^6}$.

Mean Value Theorem

States that if a function $f$ is continuous on $[-2,1]$ and differentiable on $(-2,1)$, there exists a point $c \text{ in } (-2,1)$ such that $f'(c) = \frac{f(1) - f(-2)}{1 - (-2)}$. For $f(-2) = 5$ and $f(1) = 4$, $f'(c) = -\frac{1}{3}$.

Linear Approximation

A method using the tangent line at a known point to estimate values of a function nearby, defined by $L(x) = f(a) + f'(a)(x-a)$. Using linear approximation, $e^{-0.01} \text{ is estimated as } 0.99$.

Implicit Differentiation

A technique used to find $\frac{dy}{dx}$ for an equation involving both $x$ and $y$ by differentiating both sides with respect to $x$ and using the chain rule on $y$-terms (e.g., $\frac{d}{dx}(y^4) = 4y^3 \frac{dy}{dx}$).

Logarithmic Differentiation

A technique where the natural logarithm is applied to both sides of an equation $y = f(x)$ before differentiating, useful for expressions with variable exponents such as $y = (\text{sin}(x))^{3x}$.

Related Rates

Problems where rates of change of related variables are computed using derivatives with respect to time $t$. For example, for a rectangle of area $A = l \times w$, the rate of change of area is $\frac{dA}{dt} = l \frac{dw}{dt} + w \frac{dl}{dt}$.

Critical Number

A number $c$ in the domain of a function $f$ where either the first derivative $f'(c) = 0$ or $f'(c)$ does not exist. For $f'(x) = \frac{2-5x}{(x-2)^3}$, the critical number is $x = \frac{2}{5}$.

Inflection Point

A point on the curve where the concavity changes from upward to downward or downward to upward, which occurs where $f''(x)$ changes sign. For $f''(x) = \frac{10x+4}{(x-2)^4}$, the inflection point is at $x = -\frac{2}{5}$.

Concavity

Describes the curvature of a function's graph. A graph is concave upward where $f''(x) > 0$ and concave downward where $f''(x) < 0$.