Calculus Quiz: Functions, Rates of Change, and Applications

Functions and Their Properties

Graphical Conditions and Asymptotes

This section explores the properties of functions, including limits, continuity, differentiability, and asymptotes.

• Function Value: The value of the function at a specific point, e.g., $f(1) = 3$.

• Limit: The behavior of the function as the input approaches a certain value. For example, $\lim_{x \to 0} f(x) = 1$ means as $x$ approaches 0, $f(x)$ approaches 1.

• Continuity: A function is continuous at a point if its limit at that point equals its value there.

• Differentiability: A function is differentiable at a point if its derivative exists there. If not, the graph may have a sharp corner or cusp.

• Vertical Asymptote: A line $x = a$ is a vertical asymptote if the function grows without bound as $x$ approaches $a$.

Example: Sketch a function with the following properties:

• $f(1) = 3$

• $\lim_{x \to 0} f(x) = 1$

• $\lim_{x \to 3} f(x) = 1$

• $f$ is continuous but not differentiable at $x = 3$

• Vertical asymptote at $x = 2$

Such a function might have a jump or cusp at $x = 3$, and the graph would approach infinity near $x = 2$.

Average Rate of Change

Interpreting the Average Rate of Change

The average rate of change of a function $N(x)$ over an interval $[a, b]$ is given by:

$\frac{N(b) - N(a)}{b - a}$

• Application: For $N(x) = \frac{10x + 100}{x + 2}$ on $[0, 1]$, calculate:

$N(0) = \frac{100}{2} = 50$ $N(1) = \frac{10 + 100}{1 + 2} = \frac{110}{3} \approx 36.67$ Average rate: $\frac{36.67 - 50}{1 - 0} = -13.33$

• Interpretation: The average rate of change represents how much $N(x)$ decreases per unit increase in $x$ over the interval.

Applications of Derivatives

Profit Rate of Change

Given a profit function $P(x) = 10x - 0.03x^2 - 500$, the rate at which average profit per machine changes is found by differentiating the average profit function:

• Average profit per machine: $A(x) = \frac{P(x)}{x}$

• Find $A'(x)$ at $x = 5$:

First, $A(x) = \frac{10x - 0.03x^2 - 500}{x}$ Simplify: $A(x) = 10 - 0.03x - \frac{500}{x}$ Then, $A'(x) = -0.03 + \frac{500}{x^2}$ At $x = 5$: $A'(5) = -0.03 + \frac{500}{25} = -0.03 + 20 = 19.97$

• Interpretation: When 5 machines have been sold, the average profit per machine is increasing at approximately $19.97 per machine.

Definition of the Derivative

Using the Limit Definition

The derivative of a function $p(t)$ at $t$ is defined as:

$p'(t) = \lim_{h \to 0} \frac{p(t + h) - p(t)}{h}$

• Application: For $p(t) = 272t^2 + 6000$, find $p'(t)$:

$p'(t) = \frac{d}{dt}(272t^2 + 6000) = 544t$

• Interpretation: $p'(t)$ gives the rate of change of copper consumption at time $t$.

Evaluating at Specific Years

• Find consumption in 2005: $t = 15$ (since $t = 0$ is 1990)

• $p(15) = 272 \times 15^2 + 6000 = 272 \times 225 + 6000 = 61,200 + 6000 = 67,200$

• Find rate in 2005: $p'(15) = 544 \times 15 = 8,160$

Interpretation: In 2005, copper consumption was 67,200 thousand metric tons, increasing at 8,160 thousand metric tons per year.

Motion Under Gravity

Vertical Motion Equations

The height of a stone thrown vertically is given by $h(t) = -4.9t^2 + 34t + 118$.

• Velocity: $v(t) = h'(t) = -9.8t + 34$

• Acceleration: $a(t) = v'(t) = -9.8$ (constant, due to gravity)

Evaluating at Specific Times

• Velocity at $t = 5$: $v(5) = -9.8 \times 5 + 34 = -49 + 34 = -15$ m/s

• Interpretation: Negative velocity means the stone is falling.

• Acceleration: Always $-9.8$ m/s$^2$

• Velocity at impact: Find $t$ when $h(t) = 0$; solve $-4.9t^2 + 34t + 118 = 0$ for $t$, then compute $v(t)$ at that $t$.

Example: If $t_{impact}$ is found to be approximately $-34 \pm \sqrt{34^2 - 4 \times (-4.9) \times 118} / (2 \times -4.9)$, substitute to find the exact time and velocity.

Summary Table: Key Calculus Concepts

Concept	Definition	Formula	Application
Limit	Value function approaches as input nears a point	$\lim_{x \to a} f(x)$	Continuity, asymptotes
Derivative	Instantaneous rate of change	$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$	Velocity, acceleration, rates
Average Rate of Change	Change over an interval	$\frac{f(b) - f(a)}{b - a}$	Profit, population growth
Vertical Asymptote	Line where function grows without bound	$x = a$	Graphing rational functions