Differential Calculus Review Flashcards

Flashcards on Differential Calculus covering limits, continuity types, derivative definitions, optimization principles, and geometric maxima/minima relationships.

Calculus Etymology

Derived from the Latin word "calx" meaning "stone" and from the Greek word "chalis" meaning "limestone".

Founders of Calculus

Gottfried Wilhelm von Leibniz, who published his early work on calculus in 1684, and Isaac Newton, who made an early study in 1665 and published in 1704.

Differential Calculus

A branch of mathematics which deals with derivatives and limits.

Limit of a Function

The real value $L$ that the function $f(x)$ has as the point $a$ is approached, defined by the statement $\lim_{x \to a} f(x) = L$.

Continuity Conditions at a Point

A function $f$ is continuous at a point $a$ if: 1. $f(a)$ exists, 2. $\lim_{x \to a} f(x)$ exists, and 3. $\lim_{x \to a} f(x) = f(a)$.

Jump Continuity

A type of continuity that occurs when the curve "breaks" at a particular place and starts somewhere else.

Point Discontinuity

Occurs when the curve has a "hole" in it from a missing point because the function has a value at that point that is "off the curve".

Essential Discontinuity

Occurs when the curve has a vertical asymptote.

Removable Discontinuity

Occurs when there is a rational expression with common factors in the numerator and denominator.

First Derivative

Expresses the rate of change of a function with respect to an independent variable and represents the slope of the tangent line to the curve defined by the function.

Increasing Function

A function $f$ defined on an interval $I$ where $f(x_1) < f(x_2)$ whenever $x_1, x_2$ are in $I$ and $x_1 < x_2$.

Decreasing Function

A function $f$ defined on an interval $I$ where $f(x_1) > f(x_2)$ whenever $x_1, x_2$ are in $I$ and $x_1 < x_2$.

Local Maximum Value

A value $f(c)$ on an open interval $I$ such that $f(x) \le f(c)$ for all $x$ in $I$.

Local Minimum Value

A value $f(c)$ on an open interval $I$ such that $f(x) \ge f(c)$ for all $x$ in $I$.

Steps in Solving Maxima/Minima Problems

1. Draw a figure when necessary. 2. Determine which variable (the dependent variable) is to be maximized or minimized. 3. Formulate equation. 4. Reduce to one variable. 5. Differentiate. 6. Equate to zero.

Steps in Solving Time Rates Problems

1. Draw a figure when necessary. 2. Formulate equation. 3. Differentiate with respect to time. 4. Substitute the condition / instant to the equation (substitute given values only after differentiating).

Second Derivative Test for Maxima and Minima

When the first derivative is equated to zero, a negative second derivative indicates a Maximum Point, a positive second derivative indicates a Minimum Point, and a second derivative of $0$ indicates an Inflection Point.

Largest Rectangle Inscribed in a Circle

The largest rectangle inscribed in a circle is a SQUARE.

Smallest Perimeter of a Sector with Given Area

For a sector of given area $A$, the minimum perimeter occurs when radius $r = \sqrt{A}$ and central angle $\theta = 2\,\text{rad}$.

Right Triangle with Smallest Perimeter or Largest Area

An isosceles right triangle where angle $\theta = 45^\circ$ and side lengths $x = y$.

Stiffest Beam Cut from a Circular Log

A beam cut from a circular log of radius $r$ with stiffness proportional to breath $x$ times the cube of width $y$, maximized when $y = x\sqrt{3}$.

Strongest Beam Cut from an Elliptical Section

A beam cut from an elliptical section with strength proportional to breadth $x$ times the square of depth $y$, optimized when $x = 2b\sqrt{\frac{1}{3}}$ and $y = 2a\sqrt{\frac{2}{3}}$.

Most Efficient Trapezoidal Cross-Section

The trapezoidal section with maximum capacity and minimum perimeter, formed as half of a regular hexagon with top width equal to the sum of sides ($2x$) and angle $\theta = 120^\circ$.

Maximum Length of Rigid Beam in Perpendicular Hallways

The maximum length $L$ of a rigid beam that can pass perpendicular hallways of widths $a$ and $b$ is given by $L = \left(a^{2/3} + b^{2/3}\right)^{3/2}$.

Guy Wire Ground Stake Optimization

To minimize guy wire length between two poles of heights $h_1$ and $h_2$ separated by distance $d$, the stake distance $x$ from pole $h_1$ is $x = \frac{d \cdot h_1}{h_1 + h_2}$.

Best Possible View Angle of a Picture or Clock

The distance $x$ that maximizes view angle $\theta$ for a picture or clock with bottom and top heights $y_1$ and $y_2$ is $x = \sqrt{y_1 y_2}$.

Least Amount of Material for Open Top Cylindrical Tank

An open top cylindrical tank using the least amount of material has dimensions where radius equals height, $r = h$.

Weight Ratio of Heaviest Cylinder to Circumscribing Sphere

The ratio of the weight of the heaviest cylinder to the weight of its circumscribing sphere is $\frac{W_C}{W_S} = \frac{1}{\sqrt{3}}$.