∬Differential Calculus Unit 15 – Derivatives and Graph Shape Analysis

Key Concepts and Definitions

• Derivatives measure the rate of change of a function at a given point
• The derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{d}{dx}f(x)$
  • For example, if $f(x) = x^2$, then $f'(x) = 2x$
• Instantaneous rate of change is the slope of the tangent line at a specific point on a curve
• Average rate of change is the slope of the secant line between two points on a curve
• Higher-order derivatives are derivatives of derivatives, such as $f''(x)$ (second derivative) and $f'''(x)$ (third derivative)
• Differentiability is a property of a function that ensures the existence of a derivative at every point in its domain
  • Continuous functions are not always differentiable (absolute value function at $x=0$)
• Smoothness refers to the continuity of derivatives up to a certain order

Derivative Rules and Techniques

• The power rule states that for a function $f(x) = x^n$, its derivative is $f'(x) = nx^{n-1}$
• The constant rule indicates that the derivative of a constant function is always zero
• The sum rule allows for the derivative of a sum of functions to be the sum of their individual derivatives
  • If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$
• The product rule is used to find the derivative of the product of two functions
  • If $f(x) = g(x) \cdot h(x)$, then $f'(x) = g'(x)h(x) + g(x)h'(x)$
• The quotient rule is applied to find the derivative of the quotient of two functions
  • If $f(x) = \frac{g(x)}{h(x)}$, then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$
• The chain rule is used for finding the derivative of a composite function
  • If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$
• Implicit differentiation is a technique for finding derivatives of functions defined implicitly
  • Useful for functions not expressed in the form $y = f(x)$ (circle equation $x^2 + y^2 = r^2$)

Interpreting Derivatives

• The sign of the derivative indicates whether the function is increasing (positive) or decreasing (negative) at a given point
• The magnitude of the derivative represents the rate at which the function is changing at a specific point
  • A larger magnitude implies a steeper slope and faster rate of change
• The second derivative determines the concavity of a function
  • Positive second derivative indicates concave up, negative second derivative indicates concave down
• The second derivative test helps classify critical points as local maxima, local minima, or neither
• Inflection points are points where the concavity of a function changes, and the second derivative is zero or undefined
• Higher-order derivatives provide information about the rate of change of lower-order derivatives

Graphical Analysis of Functions

• Graphing derivatives alongside the original function helps visualize the relationship between a function and its rate of change
• The derivative graph crosses the x-axis at points where the original function has horizontal tangents (critical points)
• The derivative graph is positive when the original function is increasing and negative when the original function is decreasing
• The steepness of the original function's graph is reflected in the magnitude of the derivative graph
• Inflection points of the original function correspond to critical points (x-intercepts) of the second derivative graph
• Analyzing the graphs of higher-order derivatives provides insights into the behavior of the original function and its lower-order derivatives

Critical Points and Extrema

• Critical points are points where the derivative is zero or undefined
  • Potential locations for local maxima, local minima, or inflection points
• Local maxima are points where the function value is greater than or equal to nearby points
  • First derivative is zero, and second derivative is negative
• Local minima are points where the function value is less than or equal to nearby points
  • First derivative is zero, and second derivative is positive
• Absolute (global) maximum is the highest point on a function's graph over its entire domain
• Absolute (global) minimum is the lowest point on a function's graph over its entire domain
• Fermat's theorem states that if a function has a local extremum at a point and is differentiable there, the derivative at that point must be zero
• Rolle's theorem guarantees the existence of a point with a zero derivative between any two points where a continuous function has equal values

Curve Sketching

• Begin by finding the domain of the function and any vertical or horizontal asymptotes
• Identify x- and y-intercepts by setting the function equal to zero or the input variable equal to zero
• Find critical points by setting the first derivative equal to zero and solving for x
• Determine the intervals of increase and decrease using the sign of the first derivative
• Find inflection points by setting the second derivative equal to zero and solving for x
• Determine the intervals of concavity using the sign of the second derivative
• Sketch the function by combining the information about intercepts, critical points, intervals of increase/decrease, and concavity
• Label key points and features, such as local maxima, local minima, and inflection points

Applications in Real-World Problems

• Optimization problems involve finding the maximum or minimum value of a function subject to given constraints
  • Maximizing profit, minimizing cost, or optimizing dimensions in manufacturing
• Marginal analysis in economics uses derivatives to study the effect of small changes in variables on economic outcomes
  • Marginal cost, marginal revenue, and marginal profit
• Velocity and acceleration in physics are represented by the first and second derivatives of position, respectively
• Population growth models in biology and ecology use derivatives to analyze the rate of change of populations over time
• Derivatives are used in finance to calculate the sensitivity of financial instruments to changes in underlying variables (Greeks)
  • Delta measures the rate of change of an option's price with respect to the change in the underlying asset's price

Common Pitfalls and Tips

• Remember to use the chain rule when differentiating composite functions
  • Identify the "inner" and "outer" functions and apply the chain rule accordingly
• Be careful with the order of operations when applying derivative rules, especially the product and quotient rules
• Don't forget to consider the domain of the function and any points where the derivative may be undefined
• When using the second derivative test, ensure the second derivative is not zero at the critical point
  • If the second derivative is zero, the test is inconclusive, and further analysis is needed
• Pay attention to the signs of the derivatives when determining intervals of increase/decrease and concavity
• Sketch the function step-by-step, considering all the information gathered from the derivatives and critical points
• Practice various types of problems to develop a strong understanding of the concepts and techniques involved in graph shape analysis