♾️AP Calculus AB/BC Unit 4 – Contextual Applications of Differentiation

Key Concepts

• Understand the concept of differentiation as the rate of change of a function
• Apply the power rule, product rule, quotient rule, and chain rule to differentiate functions
• Recognize the relationship between the derivative of a function and its graphical representation
• Determine critical points, inflection points, and intervals of increasing/decreasing behavior
• Utilize the first and second derivatives to analyze the behavior of a function
• Solve optimization problems by finding the maximum or minimum values of a function
• Apply the concept of related rates to solve problems involving multiple variables changing with respect to time

Real-World Applications

• Utilize differentiation to analyze the motion of objects (position, velocity, acceleration)
• Apply optimization techniques to maximize profit, minimize cost, or optimize resource allocation in business and economics
• Determine the rate of change of physical quantities in scientific applications (temperature, pressure, volume)
• Analyze the growth and decay of populations using exponential and logistic models
• Investigate the rate of change of chemical concentrations in reactions
• Optimize the design of physical structures (bridges, buildings, containers) to minimize material usage or maximize strength
• Determine the optimal dimensions of a product to minimize packaging costs or maximize volume

Graphical Interpretations

• Interpret the derivative as the slope of the tangent line to a curve at a given point
• Identify the relationship between the sign of the derivative and the increasing/decreasing behavior of a function
• Determine the concavity of a function using the second derivative
  • Concave up: second derivative is positive
  • Concave down: second derivative is negative
• Locate critical points (local maxima, local minima, and saddle points) by finding where the derivative is zero or undefined
• Analyze the behavior of a function near inflection points where the concavity changes
• Sketch the graph of a function using information from its first and second derivatives

Optimization Problems

• Identify the objective function (the quantity to be maximized or minimized) and the constraint equations
• Express the objective function in terms of a single variable using the constraint equations
• Determine the domain of the objective function based on the context of the problem
• Find the critical points of the objective function by setting its derivative equal to zero and solving for the variable
• Evaluate the objective function at the critical points and the endpoints of the domain to determine the maximum or minimum value
• Interpret the solution in the context of the original problem

Related Rates

• Recognize problems that involve multiple variables changing with respect to time
• Identify the relationship between the rates of change of the variables using the given information
• Express the rate of change of one variable in terms of the other variables and their rates of change
• Use the chain rule to differentiate the equation relating the variables
• Substitute the known values and solve for the desired rate of change
• Interpret the result in the context of the problem

Linearization and Approximation

• Understand the concept of linearization as approximating a nonlinear function with a linear function near a point
• Use the tangent line approximation formula: $L(x) = f(a) + f'(a)(x-a)$, where $a$ is the point of approximation
• Determine the accuracy of the linear approximation by comparing it to the actual value of the function
• Apply linearization to estimate the values of functions that are difficult to calculate directly (square roots, trigonometric functions)
• Recognize the limitations of linearization when the function deviates significantly from its tangent line

Common Mistakes and Pitfalls

• Forgetting to use the chain rule when differentiating composite functions
• Misidentifying the objective function or constraint equations in optimization problems
• Failing to consider the domain of the objective function based on the context of the problem
• Confusing the signs of the first and second derivatives when analyzing the behavior of a function
• Neglecting to interpret the solution in the context of the original problem
• Overrelying on linearization when the approximation is not accurate enough

Practice Problems and Solutions

1. A rectangular garden is to be enclosed by a fence. The gardener has 100 feet of fencing material. Find the dimensions of the garden that will maximize its area.

   • Let $x$ be the width and $y$ be the length of the garden
   • Constraint equation: $2x + 2y = 100$ (perimeter of the garden)
   • Objective function: $A(x) = xy = x(50-x)$ (area of the garden)
   • $A'(x) = 50 - 2x$, setting $A'(x) = 0$ gives $x = 25$
   • The dimensions that maximize the area are $x = 25$ feet and $y = 25$ feet

2. A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. Find the rate at which the radius of the balloon is increasing when the radius is 5 centimeters.

   • Let $V$ be the volume and $r$ be the radius of the balloon
   • $V = \frac{4}{3}\pi r^3$, differentiating with respect to time $t$ gives $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$
   • $\frac{dV}{dt} = 10$ cm³/s, $r = 5$ cm
   • Substituting: $10 = 4\pi (5)^2 \frac{dr}{dt}$, solving for $\frac{dr}{dt}$ gives $\frac{dr}{dt} = \frac{1}{10\pi}$ cm/s

3. Use linearization to estimate the value of $\sqrt{26}$.

   • Let $f(x) = \sqrt{x}$, choose $a = 25$ (perfect square close to 26)
   • $f(25) = 5$, $f'(x) = \frac{1}{2\sqrt{x}}$, $f'(25) = \frac{1}{10}$
   • Tangent line approximation: $L(x) = 5 + \frac{1}{10}(x-25)$
   • Estimate: $L(26) = 5 + \frac{1}{10}(26-25) = 5.1$
   • Actual value: $\sqrt{26} \approx 5.099$, the approximation is close to the actual value