Mastering Concavity & Second Derivative Test

Jan 01, 2025

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Learn to determine concave intervals, points of inflection, and find extrema using second derivative test effectively.

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Mastering Concavity & Second Derivative Test

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Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection of the graph of a function. Apply the Second Derivative Test to find relative extrema of a function.
Definition of Concavity

Ex01: Determining Concavity
Ex02: Determining Concavity
Points of Inflection
Ex03: Finding Points of Inflection Determine the points of inflection and discuss the concavity of the graph of

Ex04: Using the Second Derivative Test Find the relative extrema for
3.4 Problem Set A • 1, 4, 7 • 10, 13, 16, 19 • 25, 27 • 30, 33, 36, 39

Computation of Derivative

Derivative of a constant. In calculus, the derivative of a constant function is zero (A constant function is one that does not depend on the independent variable, such as f(x) = 7).The rule can be justified in various ways. The derivative is the slope of the tangent to the given function's graph, and the graph of a constant function is a horizontal line, whose slope is zero.ProofA formal proof, from the definition of a derivative, is:In Leibniz notation, it is written as:.

725 views • 29 slides

Parametric Equations

Parametric Equations. Eliminating the Parameter. 1). 2). 11.2 Slope and Concavity. For the curve given by Find the slope and concavity at the point (2,3). At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up. Horizontal and Vertical tangents.

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Hedging Risk

Hedging Risk. Derivative Securities. Derivative Markets. Derivative markets are a relatively new phenomenon, but are one of the most rapidly growing asset classes. Currently, there are approximately 300 million derivative contracts outstanding with a market value of around $50 Trillion

811 views • 56 slides

Learning Objectives for Section 10.4 The Derivative

Learning Objectives for Section 10.4 The Derivative . The student will be able to calculate rate of change and slope of the tangent line. The student will be able to interpret the meaning of the derivative. The student will be able to identify the nonexistence of the derivative.

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Section 3.3 – Increasing and Decreasing Functions and the First Derivative Test

Section 3.3 – Increasing and Decreasing Functions and the First Derivative Test. How Derivatives Affect the Shape of the Graph.

817 views • 26 slides

Derivative By Muhammad Shahid DA SKBZ COLLEGE

Derivative By Muhammad Shahid DA SKBZ COLLEGE. Contents. 1) What is Derivative 2)Derivative at a point 3)Example 4)Instantaneous Rate of Change 5)Application of Derivative 6)Example 7)Chain Rule 8)Example of Chain Rule 9)Higher order Derivative 10)Example of Maximum and Minimum.

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Overview

Overview. Chapter Four. Increasing and Decreasing. This is slope so you use first derivative. If it asks for intervals of increasing and decreasing f’(x)=0 to find stationary points and also find critical #s. Then make an interval table and test in the first derivative. Concavity.

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Intervals of Concavity

Unit 4. Intervals of Concavity. Definition. A graph is concave up if it forms a parabola that opens upward A graph is concave down if it forms a parabola that opens downward An inflection point is a point where a graph switches concavity. Example graphs. Concave up. Concave down .

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Overview of the Synthetic Derivative

Overview of the Synthetic Derivative. April 16, 2010 Melissa Basford, MBA Program Manager – Synthetic Derivative. Synthetic Derivative resource overview. Rich, multi-source database of de-identified clinical and demographic data Contains ~ 1.8 million records

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AP Calculus Section 3.4 Concavity and the Second Derivative Test

AP Calculus Section 3.4 Concavity and the Second Derivative Test . Student Notes Definitions, Theorems, and Examples. Definition.

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Section 3.4 – Concavity and the Second Derivative Test

Section 3.4 – Concavity and the Second Derivative Test. The Second Derivative and the Function. The first derivative tells us where a function is increasing or decreasing. But how can we tell the manner in which a function is increasing or decreasing?

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Section 4.4

Section 4.4. Concavity and the Second Derivative Test. Concavity. Concavity (cont.). Example 1. State the signs of and on the interval (0,2). Example 2. State the signs of and on the interval (0,2). Example 3.

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3.5 – Derivative of Trigonometric Functions

3.5 – Derivative of Trigonometric Functions. HW: Complete 3.5 Part 2 WKSH By the end of today you should be able to understand the derivatives of all of the trigonometric functions. Derivative of sine. Derivative of cosine. Derivative of tangent. Derivative of secant. Derivative of cosecant.

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Increasing/Decreasing Functions and Concavity

Increasing/Decreasing Functions and Concavity. Objective: Use the derivative to find where a graph is increasing/decreasing and determine concavity. Increasing/Decreasing.

1.45k views • 35 slides

Second Derivative Test

Second Derivative Test. Concavity. If the graph of a function f lies above all of its tangents, then it is called concave upward If the graph of a function f lies below all of its tangents, then it is called concave downward. Test for Concavity.

251 views • 9 slides

Section 4.4 Concavity and Second Derivative Test

Section 4.4 Concavity and Second Derivative Test. Definition. A function f ( x ) is concave up on an interval ( a,b ) if the graph of f ( x ) lies above its tangent line at each point on ( a,b ).

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Applied Calculus (MAT 121) Dr. Day Wednes day Feb 22, 2012

Applied Calculus (MAT 121) Dr. Day Wednes day Feb 22, 2012. Using the Second Derivative Is a function’s first derivative increasing or decreasing? Inflection points and concavity Increasing at an increasing rate Increasing at a decreasing rate Decreasing at an increasing rate

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5 023 MAX - Min: Optimization

5 023 MAX - Min: Optimization. AP Calculus. First Derivative Test for Max / Min TEST POINTS on either side of the critical numbers MAX :if the value changes from + to – MIN : if the value changes from – to +. Second Derivative Test for Max / Min FIND 2 nd Derivative

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5.2

The Second Derivative Test. 5.2. Find the absolute maximum and minimum. The Second Derivative Test. Classify the critical points using the Second Derivative test whenever possible. Determine the intervals on which the graph is concave up or concave down. Homework.

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QUIZ

QUIZ. QUIZ. Extrema. Inflection Pts. 1 st Derivative Test. 2nd Derivative Test. QUIZ. Graph. Graph. 3.6 A Summary of Curve Sketching. x-intercepts and y-intercepts (P.1) Symmetry (P.1) Domain and Range (P.3) Continuity (1.4) Vertical Asymptotes (1.5)

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3.1 Derivative of a Function

3.1 Derivative of a Function. In section 2.4, we defined the slope of a curve y = f(x) at the point where x = a to be When it exists, this limit is called the derivative of f at a .

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The Derivative

The Derivative. 3.1. Calculus. Differential Calc. Integral Calc. Derivative – instantaneous rate of change of one variable wrt another. Differentiation – process of finding the derivative. Finding Average Rate of Change.

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