Calculus I: Applications of the Derivative, Curve Sketching, and L'Hôpital's Rule

Mean Value Theorem and Its Applications

Mean Value Theorem (MVT)

The Mean Value Theorem is a fundamental result in calculus that connects the average rate of change of a function to its instantaneous rate of change. It provides a formal guarantee that, under certain conditions, a function will have at least one point where the tangent is parallel to the secant line joining the endpoints.

• Statement: If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that $\frac{f(b) - f(a)}{b - a} = f'(c)$

• Hypotheses: Continuity on [a, b] and differentiability on (a, b) are required.

• Conclusion: There is at least one point where the instantaneous rate of change equals the average rate of change.

Linear Approximation

Definition and Formula

Linear approximation uses the tangent line at a point to approximate the value of a function near that point. This is especially useful for estimating values of functions that are difficult to compute directly.

• Formula: For a function f differentiable at a, the linear approximation at a is $L(x) = f(a) + f'(a)(x - a)$

• Interpretation: The function L(x) is the equation of the tangent line to f at x = a.

Example: Approximating Square Roots

Suppose f(x) = \sqrt{x} and we want to approximate \sqrt{1.1} and \sqrt{0.1} using linear approximation at x = 1.

• Step 1: Compute f(1) = 1 and f'(x) = 1/(2\sqrt{x}), so f'(1) = 1/2.

• Step 2: The linear approximation is $L(x) = 1 + \frac{1}{2}(x - 1)$

• Step 3: For x = 1.1, $L(1.1) = 1 + \frac{1}{2}(1.1 - 1) = 1.05$

• Step 4: For x = 0.1, $L(0.1) = 1 + \frac{1}{2}(0.1 - 1) = 0.55$

L'Hôpital's Rule and Indeterminate Forms

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms such as 0/0 or ∞/∞. The rule states that if the limits of the numerator and denominator both approach 0 or both approach infinity, the limit of the quotient can be found by taking the limit of the quotient of their derivatives.

• Statement: If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$ (or both are ∞), and the derivatives exist near a, 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.

• Application: May need to apply the rule more than once if the resulting limit is still indeterminate.

Indeterminate Forms: 0·∞, ∞–∞, 1∞, 00, ∞0

Some limits result in indeterminate forms that require algebraic manipulation before L'Hôpital's Rule can be applied. Common strategies include rewriting products as quotients or using logarithms and exponentials.

• 0·∞ and ∞–∞: Convert to 0/0 or ∞/∞ form before applying L'Hôpital's Rule.

• 1∞, 00, ∞0: Use logarithms to bring the exponent down and convert to a product or quotient.

• Procedure: For $\lim_{x \to a} f(x)^{g(x)}$ with an indeterminate form, set $L = \lim_{x \to a} f(x)^{g(x)}$, then $\ln L = \lim_{x \to a} g(x) \ln f(x)$, and evaluate using L'Hôpital's Rule if necessary.

Absolute and Relative Extrema

Definitions and Theorems

Absolute (Global) Maximum/Minimum: The highest/lowest value of a function on a given interval. Relative (Local) Maximum/Minimum: The highest/lowest value of a function within a neighborhood of a point.

• Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on [a, b].

• Finding Extrema: Evaluate f at critical points (where f'(x) = 0 or f'(x) is undefined) and at endpoints.

Examples and Graphical Interpretation

• Critical points and endpoints must be checked to determine absolute extrema on a closed interval.

• Endpoints can be locations of absolute extrema if the function does not attain higher or lower values elsewhere.

Curve Sketching and Graphing Functions

Guidelines for Graphing

Graphing a function involves a systematic analysis of its properties using calculus. The following steps are recommended:

1. Identify the domain of the function.

2. Check for symmetry: Even, odd, or periodic behavior.

3. Find first and second derivatives: Needed for critical points, inflection points, and concavity.

4. Find critical points and possible inflection points: Where f'(x) = 0 or f'(x) is undefined.

5. Determine intervals of increase/decrease and concavity: Use the sign of f'(x) and f''(x).

6. Identify extreme values and inflection points: Use the First and Second Derivative Tests.

7. Locate asymptotes and analyze end behavior: Vertical, horizontal, or slant asymptotes.

8. Find intercepts: Where the graph crosses the axes.

9. Sketch the graph: Combine all information for an accurate sketch.

Example: Rational Function

Consider $f(x) = \frac{10x^3}{x^2 - 1}$. The graphing process involves:

• Domain: $x \neq \pm 1$

• Symmetry: Even function

• Critical points: Solve $f'(x) = 0$

• Intervals of increase/decrease and concavity: Use sign charts for $f'(x)$ and $f''(x)$

• Asymptotes: Vertical at $x = \pm 1$, slant as $x \to \infty$

• Intercepts: At $x = 0$

What Derivatives Tell Us

Intervals of Increase and Decrease

The sign of the first derivative f'(x) determines where a function is increasing or decreasing.

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

Sketching Functions Using Derivatives

By analyzing the sign of f'(x) and the points where it is zero or undefined, we can sketch the general shape of a function and identify key features such as cusps, corners, and horizontal tangents.

Summary Table: Key Calculus Concepts

Concept	Definition/Formula	Application
Mean Value Theorem	$\frac{f(b) - f(a)}{b - a} = f'(c)$	Guarantees a point with tangent parallel to secant
Linear Approximation	$L(x) = f(a) + f'(a)(x - a)$	Estimate function values near a point
L'Hôpital's Rule	$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$	Evaluate indeterminate limits
Absolute Extrema	Check critical points and endpoints	Find global max/min on closed intervals
Intervals of Increase/Decrease	Sign of $f'(x)$	Determine where function rises or falls

Additional info: These notes synthesize textbook definitions, theorems, and worked examples to provide a comprehensive review of key calculus concepts relevant to applications of the derivative, curve sketching, and L'Hôpital's Rule.