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Calculus I
Table of Contents
Unit 4 Overview: Applications of Derivatives
4.1 Related Rates
4.2 Linear Approximations and Differentials
4.3 Maxima and Minima
4.4 The Mean Value Theorem
4.5 Derivatives and the Shape of a Graph
4.6 Limits at Infinity and Asymptotes
4.7 Applied Optimization Problems
4.8 L’Hôpital’s Rule
4.9 Newton’s Method
4.10 Antiderivatives

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4.5 Derivatives and the Shape of a Graph

Citation:

Derivatives reveal a function's behavior, showing where it increases, decreases, or levels off. By analyzing the first and second derivatives, we can understand a graph's shape, including its peaks, valleys, and curves.

Applying derivative tests helps classify critical points as maxima or minima. These tools, along with concepts of continuity and differentiability, give us a powerful way to analyze functions and their graphs.

Derivatives and the Shape of a Graph

Relationship of first derivative to graph

  • The first derivative $f'(x)$ represents the rate of change or slope of the tangent line at any point on the function $f(x)$
    • Positive first derivative $f'(x) > 0$ indicates the function is increasing at that point (uphill)
    • Negative first derivative $f'(x) < 0$ indicates the function is decreasing at that point (downhill)
    • Zero first derivative $f'(x) = 0$ indicates a horizontal tangent line at that point, known as a critical point (flat)
  • The sign of the first derivative determines the monotonicity of the function over intervals
    • Positive first derivative $f'(x) > 0$ for all $x$ in an interval means the function is strictly increasing on that interval (always going up)
    • Negative first derivative $f'(x) < 0$ for all $x$ in an interval means the function is strictly decreasing on that interval (always going down)

First derivative test for extrema

  • Critical points are points where the first derivative is either zero $f'(x) = 0$ or undefined
    • Find critical points by solving $f'(x) = 0$ and identifying points where $f'(x)$ is undefined
  • The first derivative test classifies critical points as local maxima, local minima, or neither
    • First derivative changes from positive to negative at a critical point indicates a local maximum (peak)
    • First derivative changes from negative to positive at a critical point indicates a local minimum (valley)
    • No sign change in the first derivative at a critical point means neither a local maximum nor minimum (saddle point)

Second derivative and concavity

  • The second derivative $f''(x)$ represents the rate of change of the first derivative
  • The sign of the second derivative determines the concavity of the function at a point
    • Positive second derivative $f''(x) > 0$ means the function is concave up at that point (opens upward)
    • Negative second derivative $f''(x) < 0$ means the function is concave down at that point (opens downward)
  • Inflection points are points where the concavity of the function changes
    • At an inflection point, the second derivative is either zero $f''(x) = 0$ or undefined

Concavity test over intervals

  • The concavity test determines the concavity of a function over an open interval
    • Positive second derivative $f''(x) > 0$ for all $x$ in an open interval means the function is concave up on that interval (smiling curve)
    • Negative second derivative $f''(x) < 0$ for all $x$ in an open interval means the function is concave down on that interval (frowning curve)
  • Find intervals of concavity by solving inequalities $f''(x) > 0$ and $f''(x) < 0$, and identifying points where $f''(x) = 0$ or is undefined

Function behavior vs derivatives

  • The first derivative provides information about the function's increasing/decreasing behavior and critical points
  • The second derivative provides information about the function's concavity and inflection points
  • Combining information from the first and second derivatives gives a comprehensive understanding of the function's shape and behavior
    • Increasing and concave up (speeding up)
    • Increasing and concave down (slowing down)
    • Decreasing and concave up (slowing down)
    • Decreasing and concave down (speeding up)

Applying Derivative Tests

Second derivative test for extrema

  • The second derivative test is an alternative method to classify critical points as local maxima or minima
    • $f'(x) = 0$ and $f''(x) < 0$ at a critical point indicates a local maximum (peak)
    • $f'(x) = 0$ and $f''(x) > 0$ at a critical point indicates a local minimum (valley)
    • $f'(x) = 0$ and $f''(x) = 0$ at a critical point means the test is inconclusive, use the first derivative test instead
  • The second derivative test is often easier to apply than the first derivative test, as it only requires evaluating the second derivative at the critical points

Continuity, Differentiability, and Limits

  • Continuity is a prerequisite for differentiability, ensuring the function has no breaks or jumps
  • Differentiability implies that the function has a well-defined derivative at a point
  • The limit of the difference quotient as h approaches zero defines the derivative, connecting the concepts of limits and derivatives

Key Terms to Review (14)

Concave up: A function is concave up on an interval if its second derivative is positive over that interval. Graphically, this means the curve opens upwards like a cup.
Concave down: A function is concave down on an interval if its second derivative is negative on that interval. This means the graph of the function bends downward like an upside-down cup.
Concavity: Concavity describes the direction in which a curve bends. A graph is concave up if it bends upwards, and concave down if it bends downwards.
Concavity test: The concavity test determines where a function is concave up or concave down by analyzing the sign of its second derivative. If $f''(x) > 0$, the function is concave up at that point, and if $f''(x) < 0$, it is concave down.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Inflection point: An inflection point is a point on a curve where the concavity changes from concave up to concave down or vice versa. At this point, the second derivative of the function is zero or undefined.
First derivative test: The first derivative test is a method used to determine the local maxima and minima of a function by analyzing the sign changes of its first derivative. It helps in understanding the increasing and decreasing behavior of the function.
Point-slope equation: The point-slope equation of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. It is useful for writing the equation of a line when you know one point and the slope.
Second derivative test: The second derivative test helps determine the local extrema (maximum or minimum points) of a function. It uses the sign of the second derivative at critical points to assess concavity.
Slope: Slope is a measure of the steepness or incline of a line or curve, representing the rate of change between two points. It is a fundamental concept in calculus that underpins the understanding of functions, rates of change, and the shape of graphs.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
Concavity: Concavity refers to the direction in which a curve bends, indicating whether it is curving upwards or downwards. A function is concave up if its graph opens upwards like a cup, meaning that its second derivative is positive, while it is concave down if the graph opens downwards, indicating a negative second derivative. Understanding concavity is essential for analyzing the behavior of functions, particularly when it comes to identifying intervals of increase and decrease as well as determining the nature of critical points.