Critical Points and Derivative Tests

Finding Critical Values

Critical values of a function are points where the derivative is zero or undefined. These points are important for analyzing the behavior of functions, such as identifying local maxima and minima.

• Definition: A critical value of a function f(x) occurs at x = c if f'(c) = 0 or f'(c) does not exist.

• How to Find: Solve f'(x) = 0 and check where f'(x) is undefined.

• Example: For f(x) = x^3 - 3x^2 + 2, f'(x) = 3x^2 - 6x. Set 3x^2 - 6x = 0 to find critical values at x = 0 and x = 2.

First Derivative Test

The first derivative test helps determine whether a critical point is a local maximum, minimum, or neither by analyzing the sign changes of the derivative.

• Procedure: Examine the sign of f'(x) before and after each critical value.

• Interpretation:

  • If f'(x) changes from positive to negative, there is a local maximum.

  • If f'(x) changes from negative to positive, there is a local minimum.

  • If no sign change, the point is neither a max nor a min.

• Example: For f(x) = x^3 - 3x^2 + 2, check the sign of f'(x) around x = 0 and x = 2.

Second Derivative Test

The second derivative test uses the value of the second derivative at a critical point to classify it as a maximum or minimum.

• Procedure: Compute f''(c) at each critical value c where f'(c) = 0.

• Interpretation:

  • If f''(c) > 0, f has a local minimum at c.

  • If f''(c) < 0, f has a local maximum at c.

  • If f''(c) = 0, the test is inconclusive.

• Example: For f(x) = x^3 - 3x^2 + 2, f''(x) = 6x - 6.

Applications of Derivatives

Absolute Maximum and Minimum on an Interval

To find the absolute maximum and minimum values of a function on a closed interval, evaluate the function at critical points and endpoints.

• Steps:

  1. Find all critical points in the interval.

  2. Evaluate the function at each critical point and at the endpoints.

  3. The largest value is the absolute maximum; the smallest is the absolute minimum.

Intervals of Increase and Decrease

Intervals where a function increases or decreases are determined by the sign of its first derivative.

• Increasing: f'(x) > 0

• Decreasing: f'(x) < 0

• Example: For f(x) = x^3 - 3x^2 + 2, analyze f'(x) to find intervals.

Concavity and Points of Inflection

Concavity describes the direction a function curves. Points where concavity changes are called inflection points.

• Concave Up: f''(x) > 0

• Concave Down: f''(x) < 0

• Inflection Point: Where f''(x) changes sign.

Optimization in Word Problems

Derivatives are used to solve real-world problems involving maxima and minima, such as maximizing area or minimizing cost.

• Steps:

  1. Define the objective function.

  2. Find its derivative and critical points.

  3. Use tests to classify the critical points.

L'Hôpital's Rule and Indeterminate Forms

L'Hôpital's Rule

L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as 0/0 or ∞/∞.

• Statement: If \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} or \frac{\infty}{\infty}, then if the limit on the right exists.

• Example:

Antiderivatives and Integration

Finding Antiderivatives

An antiderivative of a function is a function whose derivative is the original function. The process of finding antiderivatives is called integration.

• Notation:

• Example:

Initial Value Problems

To solve for the constant of integration, use an initial condition provided in the problem.

• Example: If and , solve for .

Riemann Sums and Definite Integrals

Riemann sums approximate the area under a curve by summing areas of rectangles. The definite integral is the limit of the Riemann sum as the number of rectangles approaches infinity.

• Formula:

Fundamental Theorem of Calculus (FTC)

FTC Part 1: Area Under a Curve

The first part of the Fundamental Theorem of Calculus relates the definite integral of a function to its antiderivative.

• Statement: If F is an antiderivative of f on [a, b], then

FTC Part 2: Derivative of an Integral

The second part states that the derivative of the integral function is the original function.

• Statement: If , then

Integration Techniques

Indefinite Integrals Using Substitution

Substitution is used to simplify integrals by changing variables.

• Let , then

• Example: ; let ,

Definite Integrals Using Substitution

When using substitution in definite integrals, change the limits of integration to match the new variable.

• Example: ; let , limits become to

Integrals for Net Change and Total Flow

Integrals can be used to compute the net change or total accumulation of a quantity over an interval.

• Formula:

• Application: Used in physics for displacement, total distance, or accumulated quantity.

Area Between Curves

The area between two curves is found by integrating the difference of their functions over the interval where they intersect.

• Formula: , where on [a, b]

• Example: Find the area between and from to .