Calculus Practice Assessment 2: Derivatives, Related Rates, Optimization, and Applications

Derivatives and Differentiation Rules

Basic Derivative Computation

Calculating derivatives is a fundamental skill in calculus, allowing us to determine rates of change and slopes of tangent lines. The following problems require the use of standard derivative rules, including the power rule, product rule, chain rule, and derivatives of logarithmic and trigonometric functions.

• Power Rule: If $f(x) = x^n$, then $f'(x) = n x^{n-1}$.

• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.

• Chain Rule: If $f(x) = h(g(x))$, then $f'(x) = h'(g(x))g'(x)$.

• Derivative of $\sin x$: $\frac{d}{dx} \sin x = \cos x$.

• Derivative of $\log x$: $\frac{d}{dx} \log x = \frac{1}{x}$ (for natural logarithm).

• Derivative of $e^{f(x)}$: $\frac{d}{dx} e^{f(x)} = e^{f(x)} f'(x)$.

Examples:

• Find $\frac{d}{dx} (x^4 - 8x^3 + x)^4$ using the chain rule. I

• Find $\frac{d}{dx} \sin(\sin x)$ using the chain rule.

• Find $\frac{d}{dx} \log(e^{x} \tan x)$ using properties of logarithms and derivatives.

• Find $\frac{d}{dx} \log(\log x)$ using the chain rule.

Related Rates

Rates of Change in Geometric Contexts

Related rates problems involve finding the rate at which one quantity changes with respect to another, often in a geometric setting. These problems require implicit differentiation with respect to time.

• Volume of a Cylinder: $V = \pi r^2 h$

• Shadow and Lamppost Problem: Use similar triangles and related rates to relate the movement of a person to the rate at which the tip of the shadow moves.

• Kite Problem: Use the Pythagorean theorem to relate the string length, horizontal distance, and altitude, then differentiate with respect to time to find the rate of change of the angle.

Example:

• Given $r = 3$ (radius), $h = 5$ (height), $\frac{dr}{dt} = -1$, $\frac{dh}{dt} = 2$, find $\frac{dV}{dt}$ for a cylinder.

• A 6 ft tall person walks away from a 15 ft lamppost at 2 ft/s; find the rate at which the tip of the shadow moves when the person is 4 ft from the lamppost.

• A kite at 30 ft altitude, string let out at 2 ft/s, horizontal distance 40 ft; find $\frac{d\theta}{dt}$ where $\theta$ is the angle of inclination.

Tangent Lines and Implicit Differentiation

Finding Equations of Tangent Lines

The equation of the tangent line to a curve at a given point can be found using the derivative (slope) at that point and the point-slope form of a line.

• Point-Slope Form: $y - y_0 = m(x - x_0)$, where $m$ is the slope at $(x_0, y_0)$.

• Implicit Differentiation: Used when the equation is not solved for $y$ explicitly.

Examples:

• Find the tangent line to $x - y^2 = x^2 - 1$ at $(1, \sqrt{2})$.

• Find the tangent line to $x + \log(xy) = x + y$ at $(1,1)$.

• Find the tangent line to $y = x^2 e^x$ at $x = 1$.

• Find the tangent line to $y = g(x)$ at $x = -1$, where $g(x) = -7x^2 + 32x$.

Optimization Problems

Finding Maximum and Minimum Values

Optimization involves finding the largest or smallest value of a function, often subject to constraints. This is done by finding critical points (where the derivative is zero or undefined) and evaluating endpoints if the domain is closed.

• Critical Points: Solve $f'(x) = 0$.

• Endpoints: Evaluate $f(x)$ at the endpoints of the interval.

• Second Derivative Test: Use $f''(x)$ to determine if a critical point is a maximum or minimum.

Examples:

• Find the maximum and minimum of $\sqrt{x^2 + x}$ on $[0,1]$.

• Find the maximum and minimum of $(x^2 - 2x + 1) + \sqrt{x}$ on $[0,1]$.

• Find the maximum and minimum of $(x^2 - 2x + 1)\sqrt{x} + 4$ on $[2,4]$.

• Find the zeros and critical points of $f(x) = -2x + 2 + 2x^2$.

Cost Minimization Example:

• A rectangular pen with area 600, three sides cost $10/unit$, one side costs $30/unit$. Find dimensions minimizing cost.

Applications of Derivatives: Motion and Velocity

Projectile Motion and Average Velocity

Functions describing motion, such as the height of a firework, can be analyzed using derivatives to find velocity, maximum height, and times at which certain events occur.

• Position Function: $s(t) = -24t^2 + 96t$

• Velocity: $v(t) = s'(t)$

• Maximum Height: Occurs when $v(t) = 0$

• Average Velocity: $\frac{s(b) - s(a)}{b - a}$ over $[a, b]$

Example:

• Given $s(t) = -24t^2 + 96t$, find:

  • Height after 2 seconds

  • Velocity after 2 seconds

  • Time when velocity is zero

  • Maximum height

  • Times when firework is at ground level

  • Average velocity on $[0,2]$

  • When velocity equals average velocity on $[0,2]$

Summary Table: Key Derivative Rules

Function	Derivative
$x^n$	$n x^{n-1}$
$e^{f(x)}$	$e^{f(x)} f'(x)$
$\sin x$	$\cos x$
$\log x$	$\frac{1}{x}$
$\tan x$	$\sec^2 x$
$f(g(x))$	$f'(g(x))g'(x)$ (Chain Rule)

Additional info:

• Some function expressions in the original file were unclear; standard calculus notation and context were used to reconstruct likely intended problems.

• All problems are standard for a college-level Calculus I or II course, covering differentiation, related rates, optimization, and applications to motion.