CramX
RegisterLog in
Back to AI Flashcard MakerMathematics /AP Calculus AB: 8.2.4 The First Derivative Test

AP Calculus AB: 8.2.4 The First Derivative Test

Mathematics5 CardsCreated 3 months ago

This flashcard set explains how to use the First Derivative Test to identify relative maxima, minima, or neither at critical points. By analyzing the sign of the derivative on either side of a critical point, one can determine whether the function is increasing or decreasing, and thus classify the behavior of the function at that point.

PrintEmbedImportReport

The First Derivative Test

  • Determining the sign of the derivative immediately to either side of a critical point reveals whether that point is a relative maximum, relative minimum, or neither.
    • The first derivative test states that if c is a critical point of f, and f is continuous and differentiable on an open interval containing c (except possibly at c), then f(c) can be classified as follows:
    1) If f (c) < 0 for x < c and f (c) > 0 for x > c, then f(c) is a relative minimum of f.
    2) If f (c) > 0 for x < c and f (c) < 0 for x > c, then f(c) is a relative maximum of f.
    • On an interval, the sign of the first derivative of a function indicates whether that function is increasing or decreasing.

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/5

Key Terms

Term
Definition

The First Derivative Test

  • Determining the sign of the derivative immediately to either side of a critical point reveals whether that point is a relative maximum, rel...

note

  • Maximum and minimum values of a function will only occur at critical points: places where the derivative is zero or undefined.

  • ...

Suppose f (x) is continuous and defined for all real numbers. You are given that f (x) has critical points at x = −1 and x = 0. If f ′(x) is positive in the interval x < −1, negative in the interval −1 < x < 0, and positive in the interval x > 0, is the point where x = 0 the location of a relative maximum, minimum, or neither?

Minimum

Suppose g(x)is continuous and defined for all real numbers. You are given that g(x)has critical points at x=−5/2andx=−1. If g’(x)is positive in the interval x−1,is the point where x=−5/2 the location of a relative maximum, minimum, or neither?

Neither

Suppose that h (x) is a continuous function and is defined for all x greater than or equal to 1. You are given that h (x) has critical points at x = 1, x = 3, and x = 5. If h′ (x) is negative on the interval 1 < x < 3, positive on the interval 3 < x < 5, and positive on the interval x > 5, what can be said about the point (3, h (3))?

(3, h (3)) is an absolute minimum

Log in to view all terms

Related Flashcard Decks

Mathematics

Essential SAT Math Formulas

This deck covers key formulas and concepts essential for the SAT Math section, including slope, distance, and vertex forms.

10 cards
View Deck
Mathematics

Algebra 1 Unit 1 Mastery Test

These properties of equality help maintain balance in an equation when performing the same operation on both sides. Inverse operations, like using addition to undo subtraction, are key tools in solving equations.

29 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 2

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

53 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 1

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

55 cards
View Deck
Mathematics

Unit 2 Progress Check: MCQ Part A

To determine the point where the tangent line to the graph of the function 𝑓f has a slope of 2 for 𝑥 >0 x>0, we use the fact that the slope of the tangent line at any point is given by the derivative 𝑓′(𝑥)f′(x).

15 cards
View Deck
Mathematics

Customary Liquid Measurement Units

This set of flashcards outlines key relationships between customary units of liquid measurement, including pints, cups, quarts, and gallons. It provides conversion facts to help understand how these units relate to one another in the U.S. measurement system.

3 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 8.2.4 The First Derivative Test
TermDefinition

The First Derivative Test

  • Determining the sign of the derivative immediately to either side of a critical point reveals whether that point is a relative maximum, relative minimum, or neither.
    • The first derivative test states that if c is a critical point of f, and f is continuous and differentiable on an open interval containing c (except possibly at c), then f(c) can be classified as follows:
    1) If f (c) < 0 for x < c and f (c) > 0 for x > c, then f(c) is a relative minimum of f.
    2) If f (c) > 0 for x < c and f (c) < 0 for x > c, then f(c) is a relative maximum of f.
    • On an interval, the sign of the first derivative of a function indicates whether that function is increasing or decreasing.

note

  • Maximum and minimum values of a function will only occur at critical points: places where the derivative is zero or undefined.

  • Once you have identified the critical points, use a sign chart to illustrate the behavior of the graph on either side of each critical point.

  • The sign chart for g (x) shows that the function g is
    increasing to the left of x = –1 and decreasing to the right. Therefore, g has a maximum value at x = –1. Since the g is decreasing to the left of x = 3 and increasing to the right of it, there is a minimum value at x = 3.

  • Take a look at the possible situations around a critical point.

  • If the function changes from increasing to decreasing, it will have a relative maximum value at the critical point. If the function changes from decreasing to increasing, it will have a relative minimum at the critical point.

  • If the function does not change behavior on either side of the critical point, it will have neither a maximum value nor a minimum value at the critical point.

  • Notice the general relationship between the sign of the
    derivative, the regions where the function is increasing or decreasing, and the existence of maximum or minimum points. Using the first derivative test, you can move directly from the sign of the derivative to the existence of maxima and minima.

Suppose f (x) is continuous and defined for all real numbers. You are given that f (x) has critical points at x = −1 and x = 0. If f ′(x) is positive in the interval x < −1, negative in the interval −1 < x < 0, and positive in the interval x > 0, is the point where x = 0 the location of a relative maximum, minimum, or neither?

Minimum

Suppose g(x)is continuous and defined for all real numbers. You are given that g(x)has critical points at x=−5/2andx=−1. If g’(x)is positive in the interval x−1,is the point where x=−5/2 the location of a relative maximum, minimum, or neither?

Neither

Suppose that h (x) is a continuous function and is defined for all x greater than or equal to 1. You are given that h (x) has critical points at x = 1, x = 3, and x = 5. If h′ (x) is negative on the interval 1 < x < 3, positive on the interval 3 < x < 5, and positive on the interval x > 5, what can be said about the point (3, h (3))?

(3, h (3)) is an absolute minimum