Calculus I: Study Guide for Exam 1 (Sections 4.2–4.5, 4.8)

Mean Value Theorem and Related Concepts

Monotonicity and the Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus that connects the average rate of change of a function to its instantaneous rate of change. Understanding monotonicity and the conditions for the MVT is essential for analyzing the behavior of functions.

• Monotonicity: A function is monotonic if it is either always increasing or always decreasing on an interval.

• Strictly Increasing/Decreasing: A function f is strictly increasing if for any x_1 < x_2, f(x_1) < f(x_2). It is strictly decreasing if f(x_1) > f(x_2).

• Relationship to Derivative: If f'(x) > 0 on an interval, f is increasing there. If f'(x) < 0, f is decreasing.

• Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that $f'(c) = \frac{f(b) - f(a)}{b - a}$

• Rolle's Theorem: A special case of the MVT where f(a) = f(b). Then there exists c in (a, b) such that f'(c) = 0.

• Checking Assumptions: To apply MVT or Rolle's Theorem, verify continuity on [a, b] and differentiability on (a, b).

• Counterexamples: If assumptions are not met (e.g., function not continuous or not differentiable), the conclusion of the theorem does not hold.

• Implications: If f'(x) = 0 for all x in an interval, then f is constant on that interval. If two functions have the same derivative, they differ by a constant.

Example: For f(x) = x^2 on [1, 3], MVT guarantees a c such that f'(c) = (f(3) - f(1))/(3 - 1) = (9 - 1)/2 = 4. Since f'(x) = 2x, 2c = 4 so c = 2.

Monotonic Functions and the First Derivative Test

Local Extrema and Critical Points

Understanding how to find and classify local maxima and minima is crucial for analyzing the shape of a function's graph.

• Local Extrema: Points where a function reaches a local maximum or minimum value.

• 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 by analyzing the sign of f'(x) before and after the point.

• Endpoints: For closed intervals, endpoints must also be checked for extrema.

• Graphical Analysis: Local extrema can be identified from the graph of f or f'.

Example: For f(x) = x^3 - 3x, f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = \pm 1. Use the first derivative test to classify these points.

Concavity, Inflection Points, and Curve Sketching

Concavity and the Second Derivative Test

Concavity describes how a function bends, and the second derivative provides information about this property. Inflection points and asymptotes are also important for graphing functions.

• Concavity: If f''(x) > 0, the graph is concave up (shaped like a cup). If f''(x) < 0, it is concave down (shaped like a cap).

• Second Derivative Test: At a critical point c where f'(c) = 0:

  • 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.

• Inflection Points: Points where the concavity changes. Candidates occur where f''(x) = 0 or f''(x) does not exist. Confirm by checking the sign of f''(x) on either side.

• Curve Sketching: Use the signs of f, f', and f'' to determine intervals of increase/decrease and concavity.

• Asymptotes:

  • Vertical Asymptotes: Where the function approaches infinity as x approaches a certain value.

  • Horizontal Asymptotes: Where the function approaches a constant value as x approaches infinity or negative infinity.

Example: For f(x) = x^3, f''(x) = 6x. The inflection point is at x = 0 since concavity changes from down to up.

Indeterminate Forms and L'Hôpital's Rule

Evaluating Limits with L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms.

• Indeterminate Forms: Common forms include $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty^0$, $0^0$, $1^\infty$.

• L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ provided the limit on the right exists.

• Other Indeterminate Forms: For $0 \cdot \infty$, rewrite as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ by algebraic manipulation.

• Exponential Indeterminate Forms: For $0^0$, $1^\infty$, or $\infty^0$, use logarithms and rewrite the limit in terms of $e^{\lim}$, then apply L'Hôpital's Rule to the exponent.

Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (using L'Hôpital's Rule: $\frac{\cos x}{1}$ at $x=0$ gives 1).

Antiderivatives and Indefinite Integrals

Finding Antiderivatives

Antiderivatives are the reverse process of differentiation. The set of all antiderivatives of a function is called the indefinite integral.

• Antiderivative: A function F(x) is an antiderivative of f(x) if F'(x) = f(x).

• Indefinite Integral: The collection of all antiderivatives of f(x) is denoted by $\int f(x)\,dx = F(x) + C$, where C is an arbitrary constant.

• Finding Antiderivatives: Use known rules and patterns to integrate basic functions.

• General Solution: All antiderivatives of a function differ by a constant.

Example: $\int x^2\,dx = \frac{1}{3}x^3 + C$