Derivatives are powerful tools for analyzing functions and solving real-world problems. They help us understand how things change, find optimal solutions, and make predictions. From finding slopes to optimizing profits, derivatives are the calculus equivalent of a Swiss Army knife.

In this section, we'll explore how derivatives can be used to analyze function behavior, solve optimization problems, and calculate rates of change. We'll also dive into tangent lines and linear approximations, showing how these concepts connect to the broader world of calculus.

Function Behavior with Derivatives

Analyzing Functions with First and Second Derivatives

• The first derivative of a function represents the rate of change or slope of the tangent line at any point on the function's graph
  • Determines where the function is increasing (positive first derivative) or decreasing (negative first derivative)
• The second derivative of a function represents the rate of change of the first derivative
  • Determines the concavity of the function's graph (concave up if second derivative is positive, concave down if negative)
  • Locates inflection points where the concavity changes (second derivative equals zero)

Critical Points and Extrema

• Critical points occur where the first derivative is zero or undefined
  • Can be classified as local maxima, local minima, or neither, based on the behavior of the function around the critical point
• The First Derivative Test uses the sign of the first derivative on either side of a critical point to determine if the point is a local maximum, local minimum, or neither
  • If the first derivative changes from positive to negative, the point is a local maximum
  • If the first derivative changes from negative to positive, the point is a local minimum
  • If the first derivative does not change sign, the point is neither a maximum nor a minimum
• The Second Derivative Test uses the sign of the second derivative at a critical point to determine if the point is a local maximum, local minimum, or neither
  • If the second derivative is negative, the point is a local maximum
  • If the second derivative is positive, the point is a local minimum
  • If the second derivative is zero, the test is inconclusive
• Absolute (global) maxima and minima are the highest and lowest points on a function's graph within a given domain
  • Can occur at critical points or at the endpoints of a closed interval

Optimization with Derivatives

Setting Up Optimization Problems

• Optimization problems involve finding the maximum or minimum value of a function within given constraints
• The objective function is the function that needs to be maximized or minimized
  • Example: In a problem about maximizing profit, the profit function would be the objective function
• The constraints are the limitations or conditions that must be satisfied
  • Example: In a problem about minimizing material cost for a container with a fixed volume, the volume equation would be a constraint

Solving Optimization Problems

• To solve an optimization problem, first identify the objective function and the constraints
• Express the objective function in terms of a single variable using the constraints
  • Example: If the problem involves a rectangular area and one side length is given in terms of the other, substitute the constraint into the area function
• Find the critical points of the objective function by setting its first derivative equal to zero and solving for the variable
• Evaluate the objective function at each critical point and at the endpoints of the domain (if applicable) to determine the absolute maximum or minimum value
• In some cases, the second derivative test can be used to confirm the nature of the critical points (maximum or minimum)

Derivatives for Rates of Change

Instantaneous Rates of Change and Marginal Analysis

• Rates of change describe how one quantity changes with respect to another
  • Derivatives can be used to find instantaneous rates of change at a specific point
• Marginal analysis involves using derivatives to determine the change in one quantity resulting from a small change in another quantity
  • Marginal cost is the change in total cost resulting from producing one additional unit
  • Marginal revenue is the change in total revenue resulting from selling one additional unit
  • Example: If the cost function is $C(x) = 100 + 5x + 0.1x^2$, the marginal cost at $x = 10$ units is $C'(10) = 5 + 0.2(10) = 7$

Related Rates and Motion

• Related rates problems involve finding the rate of change of one quantity given the rate of change of another quantity
  • Express the relationship between the quantities using an equation and differentiate with respect to time
  • Example: If the radius of a circle is increasing at a rate of 2 cm/s, the area of the circle is increasing at a rate of $\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 4\pi r$ cm²/s
• Velocity is the rate of change of position with respect to time, while acceleration is the rate of change of velocity with respect to time
  • These concepts can be understood using derivatives
  • Example: If the position function is $s(t) = 3t^2 + 2t$, the velocity at time $t$ is $v(t) = s'(t) = 6t + 2$ and the acceleration is $a(t) = v'(t) = 6$

Functions vs Tangent Lines

Tangent Lines and Their Equations

• The tangent line to a function at a given point is a straight line that touches the function at that point and has the same slope as the function at that point
• The slope of the tangent line at a point is equal to the value of the function's first derivative at that point
• The equation of the tangent line to a function $f(x)$ at a point $(a, f(a))$ is given by: $y - f(a) = f'(a)(x - a)$, where $f'(a)$ is the value of the first derivative at $x = a$
  • Example: If $f(x) = x^2$ and $a = 1$, then $f(1) = 1$ and $f'(1) = 2$, so the tangent line equation is $y - 1 = 2(x - 1)$ or $y = 2x - 1$

Linear Approximation

• Linear approximation (or tangent line approximation) uses the tangent line at a point to estimate the value of the function near that point
  • The approximation is given by: $f(x) ≈ f(a) + f'(a)(x - a)$
  • Example: If $f(x) = \sqrt{x}$ and we want to estimate $\sqrt{10}$, we can use the tangent line at $a = 9$. With $f(9) = 3$ and $f'(9) = \frac{1}{6}$, the approximation is $\sqrt{10} ≈ 3 + \frac{1}{6}(10 - 9) = 3.17$
• The accuracy of the linear approximation depends on the proximity of $x$ to $a$ and the behavior of the function around that point
  • The approximation is more accurate when $x$ is closer to $a$ and when the function is nearly linear around $a$
  • Example: The linear approximation of $\sqrt{9.1}$ using the tangent line at $a = 9$ would be more accurate than the approximation of $\sqrt{10}$ because 9.1 is closer to 9