Precalculus Study Guide: Derivatives, Extrema, and Applications

Derivatives and Their Applications

Basic Differentiation

Differentiation is a fundamental concept in calculus, used to determine the rate at which a function changes. The derivative of a function at a point gives the slope of the tangent line at that point.

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

• Constant Multiple Rule: If $f(x) = a g(x)$, then $f'(x) = a g'(x)$.

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

Example: Find $\frac{d}{dx}(5x^4 + 2x^2)$.

• Apply the power rule to each term:

• $\frac{d}{dx}(5x^4) = 20x^3$

• $\frac{d}{dx}(2x^2) = 4x$

• So, $\frac{d}{dx}(5x^4 + 2x^2) = 20x^3 + 4x$

Applications of Derivatives

Derivatives are used to find rates of change, such as velocity, and to solve optimization problems.

• Rate of Change: If $P(t)$ is a population at time $t$, then $P'(t)$ is the rate of change of the population.

• Example: If $P(t) = 7t^2 + 11$, then $P'(t) = 14t$.

• To find the rate of change at $t = 5$, substitute $t = 5$ into $P'(t)$: $P'(5) = 14 \times 5 = 70$.

Extrema and Critical Points

Local and Absolute Extrema

Extrema refer to the maximum and minimum values of a function. Local extrema are the highest or lowest points in a small neighborhood, while absolute extrema are the highest or lowest points over the entire domain.

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ does not exist.

• First Derivative Test: Used to determine if a critical point is a local maximum, minimum, or neither.

• Second Derivative Test: If $f''(x) > 0$ at a critical point, it is a local minimum; if $f''(x) < 0$, it is a local maximum.

Example: For $f(x) = x^3 - 6x^2 + 12x - 3$, find local extrema.

• Find $f'(x) = 3x^2 - 12x + 12$.

• Set $f'(x) = 0$: $3x^2 - 12x + 12 = 0$.

• Solve for $x$ to find critical points.

Intervals of Increase and Decrease

A function is increasing where its derivative is positive and decreasing where its derivative is negative.

• Find intervals by solving $f'(x) > 0$ (increasing) and $f'(x) < 0$ (decreasing).

Concavity and Points of Inflection

Concavity

Concavity describes the direction a curve bends. A function is concave upward where $f''(x) > 0$ and concave downward where $f''(x) < 0$.

• Point of Inflection: A point where the concavity changes (i.e., $f''(x) = 0$ and changes sign).

Example: For $f(x) = x^3 - 12x^2 + 63x + 51$, find points of inflection.

• Find $f''(x) = 6x - 24$.

• Set $f''(x) = 0$: $6x - 24 = 0 \implies x = 4$.

• Check sign change around $x = 4$ to confirm inflection point.

Optimization Problems

Maximizing or Minimizing Quantities

Optimization involves finding the maximum or minimum values of a function, often subject to constraints.

• Revenue Maximization: If $p = 60 - 2x$, revenue $R = p \cdot x$.

• Express $R$ in terms of $x$, differentiate, set $R'(x) = 0$, and solve for $x$ to find the value that maximizes revenue.

• Area and Perimeter Problems: For a rectangle with area $A$ and sides $x$ and $y$, $A = x y$. To minimize perimeter $P = 2x + 2y$ for a fixed area, use $y = A/x$ and minimize $P(x) = 2x + 2(A/x)$.

Tables: Summary of Extrema and Concavity Tests

Test	Purpose	How to Apply
First Derivative Test	Find local maxima/minima	Check sign of $f'(x)$ before and after critical points
Second Derivative Test	Classify critical points	If $f''(x) > 0$, local min; if $f''(x) < 0$, local max
Concavity Test	Find intervals of concavity	Check sign of $f''(x)$ on intervals
Inflection Point Test	Find points where concavity changes	Set $f''(x) = 0$ and check sign change

Graphical Analysis

Interpreting Graphs of Functions and Their Derivatives

Graphs can be used to identify critical points, intervals of increase/decrease, and points of inflection.

• Critical points correspond to where the derivative graph crosses the x-axis.

• Intervals where the derivative is positive indicate the function is increasing.

• Points where the second derivative changes sign are inflection points.

Summary

• Use derivatives to find rates of change, extrema, and solve optimization problems.

• Apply the first and second derivative tests to classify critical points.

• Analyze concavity and inflection points using the second derivative.

• Optimization problems often require expressing one variable in terms of another and differentiating.