🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 8 – Differentiation & Mean Value Theorem

Key Concepts and Definitions

• Differentiation the process of finding the derivative of a function, which measures the rate of change of the function at a given point
• Derivative the slope of the tangent line to a function at a specific point, denoted as $f'(x)$ for a function $f(x)$
• Tangent line a straight line that touches a curve at a single point without crossing it
• Continuity a function is continuous if it has no breaks, gaps, or jumps in its graph
  • Continuous functions can be differentiated at every point in their domain
• Differentiability a function is differentiable at a point if its derivative exists at that point
  • Differentiability implies continuity, but continuity does not imply differentiability (e.g., $f(x) = |x|$ is continuous but not differentiable at $x=0$)
• Limit a value that a function approaches as the input approaches a specific value, denoted as $\lim_{x \to a} f(x) = L$
• Instantaneous rate of change the rate of change of a function at a specific instant or point, given by the derivative

Differentiation Basics

• The derivative of a function $f(x)$ is denoted as $f'(x)$, and it represents the rate of change of $f(x)$ with respect to $x$
• The derivative can be found using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Derivatives can be used to find the slope of a tangent line to a function at a given point
  • The equation of the tangent line to $f(x)$ at the point $(a, f(a))$ is $y - f(a) = f'(a)(x - a)$
• Higher-order derivatives can be found by differentiating the function multiple times
  • The second derivative, denoted as $f''(x)$, is the derivative of the first derivative
  • The third derivative, denoted as $f'''(x)$, is the derivative of the second derivative, and so on
• The derivative of a constant function is always 0, as the rate of change is 0 for all values of $x$
• The derivative of a linear function $f(x) = mx + b$ is the constant $m$, which represents the slope of the line
• The derivative of a polynomial function can be found by applying the power rule and adding the results

Rules and Techniques of Differentiation

• Power Rule for functions of the form $f(x) = x^n$, the derivative is $f'(x) = nx^{n-1}$
• Constant Multiple Rule if $f(x)$ is a function and $c$ is a constant, then $\frac{d}{dx}[cf(x)] = cf'(x)$
• Sum Rule if $f(x)$ and $g(x)$ are differentiable functions, then $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
• Difference Rule if $f(x)$ and $g(x)$ are differentiable functions, then $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$
• Product Rule if $f(x)$ and $g(x)$ are differentiable functions, then $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
• Quotient Rule if $f(x)$ and $g(x)$ are differentiable functions, then $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$, provided $g(x) \neq 0$
• Chain Rule if $y = f(u)$ and $u = g(x)$ are differentiable functions, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
  • The Chain Rule allows for the differentiation of composite functions

Mean Value Theorem

• The Mean Value Theorem (MVT) 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, the MVT implies 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 MVT is a powerful tool for proving various theorems in calculus, such as the Intermediate Value Theorem and Rolle's Theorem
  • Rolle's Theorem is a special case of the MVT where $f(a) = f(b)$, implying that there exists a point $c$ in $(a, b)$ such that $f'(c) = 0$
• The MVT can be used to find the average rate of change of a function over an interval
• The MVT is essential for understanding the behavior of functions and their derivatives
• The MVT can be used to prove the Fundamental Theorem of Calculus, which relates differentiation and integration

Applications of Differentiation

• Optimization finding the maximum or minimum values of a function within given constraints
  • Example: Minimizing the cost of production while maximizing profit
• Related Rates determining the rate at which one quantity changes with respect to another related quantity
  • Example: Finding the rate at which the water level in a conical tank changes as water is pumped out at a constant rate
• Marginal Analysis studying the effect of a small change in one variable on another variable
  • Example: Marginal cost is the change in total cost when producing one additional unit of a product
• Approximation using derivatives to estimate the value of a function near a given point
  • Example: Using linear approximation (tangent line) to estimate the value of a function at a point close to a known value
• Newton's Method an iterative algorithm for finding the roots (zeros) of a function using the derivative
  • The method uses the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ to successively approximate the root
• Curve Sketching using derivatives to analyze the behavior of a function and sketch its graph
  • First derivative test: Increasing (f'(x) > 0) and decreasing (f'(x) < 0) intervals
  • Second derivative test: Concavity (f''(x) > 0 for concave up, f''(x) < 0 for concave down) and inflection points (f''(x) = 0)

Graphical Interpretations

• The derivative of a function at a point is the slope of the tangent line to the function at that point
• The sign of the derivative indicates whether the function is increasing (f'(x) > 0), decreasing (f'(x) < 0), or has a horizontal tangent (f'(x) = 0) at a given point
• The second derivative of a function determines the concavity of the function's graph
  • If f''(x) > 0, the graph is concave up (opening upward)
  • If f''(x) < 0, the graph is concave down (opening downward)
  • If f''(x) = 0, the point is an inflection point, where the concavity changes
• Local maxima and minima can be identified using the first and second derivative tests
  • Local maximum: f'(x) = 0 and f''(x) < 0
  • Local minimum: f'(x) = 0 and f''(x) > 0
• The Mean Value Theorem can be visualized as a point on the curve where the tangent line is parallel to the secant line connecting the endpoints of the interval
• Graphical interpretations of derivatives help in understanding the behavior and properties of functions

Common Challenges and Misconceptions

• Confusing the concepts of continuity and differentiability
  • A function can be continuous but not differentiable at a point (e.g., absolute value function at x=0)
• Misapplying differentiation rules, especially the Chain Rule and Quotient Rule
  • It's essential to identify the correct rule for each situation and apply it properly
• Forgetting to use the Chain Rule when differentiating composite functions
  • The Chain Rule is necessary when a function is composed of other functions
• Misinterpreting the meaning of the derivative at a point
  • The derivative represents the instantaneous rate of change, not the average rate of change
• Incorrectly identifying critical points and inflection points
  • Critical points are where the derivative is zero or undefined, while inflection points are where the concavity changes
• Misunderstanding the relationship between the Mean Value Theorem and Rolle's Theorem
  • Rolle's Theorem is a special case of the MVT where the function values at the endpoints are equal
• Struggling with the concept of related rates and setting up the appropriate equations
  • It's crucial to identify the relationship between the changing quantities and use the Chain Rule to solve related rates problems

Practice Problems and Examples

1. Find the derivative of $f(x) = x^3 - 2x^2 + 4x - 1$
2. Find the equation of the tangent line to the curve $y = x^2 - 3x + 1$ at the point $(2, -3)$
3. Use the Chain Rule to find the derivative of $f(x) = (3x^2 - 2x + 1)^4$
4. Find the critical points and inflection points of the function $f(x) = x^3 - 3x^2 - 9x + 5$
5. Verify that the function $f(x) = x^3 - x$ satisfies the conditions of the Mean Value Theorem on the interval $[-1, 1]$, and find the value of $c$ guaranteed by the theorem
6. A ladder 10 feet long is leaning against a wall. If the bottom of the ladder is sliding away from the wall at a rate of 2 feet per second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?
7. Sketch the graph of the function $f(x) = x^3 - 3x^2 + 2$, labeling the critical points, inflection points, and intervals of increasing/decreasing and concavity
8. Find the absolute maximum and minimum values of the function $f(x) = x^3 - 3x^2 - 9x + 7$ on the interval $[-2, 4]$