∫Calculus I Unit 4 – Applications of Derivatives

Key Concepts and Definitions

• Derivative represents the instantaneous rate of change of a function at a specific point
• Tangent line touches a curve at a single point and provides the best linear approximation to the curve at that point
• Local extrema are the highest or lowest points of a function within a given interval
  • Local maximum is a point where the function value is greater than or equal to the function values at nearby points
  • Local minimum is a point where the function value is less than or equal to the function values at nearby points
• Global extrema are the highest (global maximum) or lowest (global minimum) points of a function over its entire domain
• Critical points are points where the derivative of a function is either zero or undefined
• Inflection points are points where the concavity of a function changes (from concave up to concave down, or vice versa)
• Optimization involves finding the maximum or minimum values of a function subject to given constraints
• Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another quantity

Rate of Change and Tangent Lines

• The average rate of change of a function $f(x)$ over the interval $[a, b]$ is given by $\frac{f(b) - f(a)}{b - a}$
• The instantaneous rate of change of a function $f(x)$ at a point $x = a$ is given by the limit of the average rate of change as the interval approaches zero: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
• The derivative of a function $f(x)$ at a point $x = a$ is denoted by $f'(a)$ and represents the instantaneous rate of change at that point
• The equation of the tangent line to a curve $y = f(x)$ at the point $(a, f(a))$ is given by $y - f(a) = f'(a)(x - a)$
  • The slope of the tangent line is equal to the value of the derivative at the point of tangency
• Tangent line approximation can be used to estimate the value of a function near a given point
  • For small changes in $x$, the change in $y$ can be approximated using the tangent line: $\Delta y \approx f'(a) \Delta x$
• Higher-order derivatives (second derivative $f''(x)$, third derivative $f'''(x)$, etc.) provide information about the rate of change of the rate of change

Finding Extrema: Local and Global

• To find local extrema, first find the critical points of the function by setting the first derivative equal to zero or identifying points where the derivative is undefined
• Evaluate the function at each critical point and at the endpoints of the domain (if the domain is a closed interval) to find the local maximum and minimum values
• To determine the nature of a critical point (local maximum, local minimum, or neither), use the first derivative test or the second derivative test
  • First derivative test: If the derivative changes sign from positive to negative at a critical point, it is a local maximum. If the derivative changes sign from negative to positive, it is a local minimum. If the derivative does not change sign, the point is neither a local maximum nor a local minimum
  • Second derivative test: If the second derivative is negative at a critical point, it is a local maximum. If the second derivative is positive, it is a local minimum. If the second derivative is zero, the test is inconclusive
• To find global extrema, compare the function values at all local extrema and the endpoints of the domain (if applicable)
  • The global maximum is the largest function value among all local maxima and endpoints
  • The global minimum is the smallest function value among all local minima and endpoints

Optimization Problems

• Optimization problems involve finding the maximum or minimum value of a function subject to given constraints
• Steps to solve optimization problems:
  1. Identify the objective function (the quantity to be maximized or minimized)
  2. Identify the constraints and express them as equations or inequalities
  3. Use the constraints to express the objective function in terms of a single variable
  4. Find the domain of the objective function based on the constraints
  5. Find the critical points of the objective function within the domain
  6. Evaluate the objective function at the critical points and the endpoints of the domain (if applicable) to determine the maximum or minimum value
• Common types of optimization problems include:
  • Maximizing area or volume subject to perimeter or surface area constraints
  • Minimizing cost or time subject to production or distance constraints
  • Optimizing revenue or profit based on demand and cost functions
• It is essential to interpret the results in the context of the original problem and ensure that the solution is feasible and practical

Related Rates

• Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another quantity
• The quantities in related rates problems are often related by geometric formulas or equations
• Steps to solve related rates problems:
  1. Identify the quantities that are changing with respect to time and assign variables to them
  2. Write an equation that relates the quantities based on the given information
  3. Differentiate both sides of the equation with respect to time, using the chain rule if necessary
  4. Substitute the known values and solve for the desired rate of change
• Common types of related rates problems include:
  • Rates of change of the volume of a shape as its dimensions change (cone, sphere, cylinder)
  • Rates of change of the distance between moving objects (boats, cars, airplanes)
  • Rates of change of the angle between lines or the length of a shadow as the position of an object changes
• It is important to pay attention to units and ensure that the rates of change are consistent (e.g., meters per second, radians per minute)

Mean Value Theorem

• The Mean Value Theorem states that if a function $f(x)$ 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 $f'(c) = \frac{f(b) - f(a)}{b - a}$
  • Geometrically, this means that there is a point on the curve where the tangent line is parallel to the secant line connecting the endpoints of the interval
• The Mean Value Theorem is a consequence of Rolle's Theorem, which states that if a function $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists a point $c$ in $(a, b)$ such that $f'(c) = 0$
• The Mean Value Theorem has several important applications:
  • Proving that a function has at most one root (zero) in an interval
  • Establishing inequalities involving the derivative of a function
  • Proving the Fundamental Theorem of Calculus
• Cauchy's Mean Value Theorem is a generalization of the Mean Value Theorem for two functions, stating that if $f(x)$ and $g(x)$ are continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a point $c$ in $(a, b)$ such that $[f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c)$

L'Hôpital's Rule

• L'Hôpital's Rule is a technique for evaluating limits of indeterminate forms, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$
• If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is an indeterminate form of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and the limit $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$
• L'Hôpital's Rule can be applied repeatedly if the resulting limit is still an indeterminate form
• Other indeterminate forms, such as $0 \cdot \infty$, $\infty - \infty$, $0^0$, $\infty^0$, and $1^\infty$, can be transformed into $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms by algebraic manipulation before applying L'Hôpital's Rule
• L'Hôpital's Rule is particularly useful for evaluating limits involving exponential, logarithmic, and trigonometric functions
• It is important to verify that the hypotheses of L'Hôpital's Rule are satisfied before applying it, and to check that the resulting limit exists

Curve Sketching

• Curve sketching is the process of creating a rough graph of a function using information about its key features, such as:
  • Domain and range
  • Intercepts (x-intercepts and y-intercept)
  • Symmetry (even, odd, or periodic)
  • Asymptotes (vertical, horizontal, or oblique)
  • Intervals of increase and decrease
  • Local and global extrema
  • Concavity and inflection points
• Steps to sketch a curve:
  1. Determine the domain of the function
  2. Find the intercepts by setting $x = 0$ and $y = 0$
  3. Check for symmetry by examining the function's equation
  4. Find the vertical asymptotes by setting the denominator equal to zero (for rational functions)
  5. Find the horizontal asymptote by evaluating the limits as $x$ approaches $\infty$ and $-\infty$
  6. Find the intervals of increase and decrease by solving $f'(x) > 0$ and $f'(x) < 0$
  7. Find the local extrema by solving $f'(x) = 0$ and classifying the critical points
  8. Find the intervals of concavity and inflection points by solving $f''(x) > 0$, $f''(x) < 0$, and $f''(x) = 0$
  9. Plot the key points and features, and connect them with a smooth curve
• Curve sketching is a valuable tool for understanding the behavior of a function and visualizing its graph without relying on technology
• Practice and experience with a variety of functions can help develop intuition and proficiency in curve sketching