Calculus Study Guide: Mean Value Theorem, Monotonicity, Curve Sketching, L'Hôpital's Rule, and Antiderivatives

Section 4.2: The Mean Value Theorem

Monotonicity and Its Implications

The concept of monotonicity describes whether a function is consistently increasing or decreasing over an interval.

• Strictly Increasing Function: For all $x_1 < x_2$ in the interval, $f(x_1) < f(x_2)$.

• Strictly Decreasing Function: For all $x_1 < x_2$, $f(x_1) > f(x_2)$.

• The sign of the derivative $f'(x)$ indicates monotonicity:

  • If $f'(x) > 0$ on an interval, $f$ is increasing there.

  • If $f'(x) < 0$, $f$ is decreasing.

Rolle’s Theorem and the Mean Value Theorem (MVT)

• Rolle’s Theorem: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists $c \in (a, b)$ such that $f'(c) = 0$.

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

• To apply these theorems, check the function’s continuity and differentiability on the required intervals.

• Counterexamples: If a function is not continuous or not differentiable, the theorems do not apply. For example, $f(x) = |x|$ on $[-1, 1]$ is not differentiable at $x=0$.

• Finding the Point $c$: Set $f'(c)$ equal to the average rate of change and solve for $c$.

• 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 $[1, 3]$, $f'(x) = 2x$. The average rate of change is $(9-1)/(3-1) = 4$. Set $2c = 4 \implies c = 2$.

Section 4.3: Monotonic Functions and the First Derivative Test

Local Extrema and Critical Points

Local extrema are points where a function reaches a local maximum or minimum.

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

• To find local extrema:

  • Find all critical points and endpoints of the interval.

  • Evaluate $f$ at these points to determine maxima and minima.

The First Derivative Test

• If $f'(x)$ changes from positive to negative at a critical point, $f$ has a local maximum there.

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

• If $f'(x)$ does not change sign, the point is not a local extremum.

Example: For $f(x) = x^3 - 3x$, $f'(x) = 3x^2 - 3$. Setting $f'(x) = 0$ gives $x = \pm 1$. Analyze sign changes to classify extrema.

Section 4.4: Concavity and Curve Sketching

Concavity and the Second Derivative

Concavity describes the direction a curve bends.

• If $f''(x) > 0$, the graph is concave up (shaped like a cup).

• If $f''(x) < 0$, the graph is concave down (shaped like a cap).

Second Derivative Test for Local Extrema

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

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

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

Points of Inflection

• A point of inflection is where the concavity changes.

• To find inflection points:

  • Find where $f''(x) = 0$ or $f''(x)$ does not exist.

  • Check that $f''(x)$ changes sign at these points.

Curve Sketching

• Determine the sign of $f(x)$, $f'(x)$, and $f''(x)$ to analyze the function’s behavior.

• Identify intervals of increase/decrease and concavity.

• Locate asymptotes:

  • Vertical asymptotes: Where $f(x)$ approaches infinity as $x$ approaches a value.

  • Horizontal asymptotes: The value $f(x)$ approaches as $x \to \infty$ or $x \to -\infty$.

Example: For $f(x) = x^3$, $f''(x) = 6x$. $f''(0) = 0$ and $f''(x)$ changes sign at $x=0$, so $(0,0)$ is an inflection point.

Section 4.5: Intermediate Forms and L’Hôpital’s Rule

Indeterminate Forms

Some limits cannot be directly evaluated and are called indeterminate forms. Common types include:

• $\frac{0}{0}$

• $\frac{\infty}{\infty}$

• $0 \cdot \infty$

• $\infty - \infty$

• $0^0$

• $1^\infty$

• $\infty^0$

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.

• For $0 \cdot \infty$ forms, rewrite as a quotient (e.g., $f(x) \cdot g(x) = \frac{f(x)}{1/g(x)}$).

• For $0^0$, $1^\infty$, or $\infty^0$, take logarithms and rewrite the limit in terms of exponentials and apply L’Hôpital’s Rule to the exponent.

Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (both numerator and denominator approach 0, so apply L’Hôpital’s Rule).

Section 4.8: Antiderivatives

Finding Antiderivatives

An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.

• The indefinite integral of $f(x)$ is the set of all antiderivatives: $$ \int f(x)\,dx = F(x) + C$$where$C$ is an arbitrary constant.

• All antiderivatives of a function differ by a constant.

• To find the indefinite integral, use known integration formulas and rules (e.g., power rule, substitution).

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