Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Concavity

Calculus

5. Graphical Applications of Derivatives

Concavity

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1:52 minutes

Problem 4.4.114b

Textbook Question

114. Parabolas
b. When is the parabola concave up? Concave down?

Verified step by step guidance

To determine when a parabola is concave up or concave down, we need to look at the second derivative of its equation. A parabola is typically given by the quadratic function \( f(x) = ax^2 + bx + c \).

The second derivative of the function \( f(x) = ax^2 + bx + c \) is \( f''(x) = 2a \). This is a constant value, meaning it does not depend on \( x \).

A parabola is concave up when the second derivative is positive. Therefore, if \( 2a > 0 \), the parabola is concave up. This simplifies to \( a > 0 \).

Conversely, a parabola is concave down when the second derivative is negative. Therefore, if \( 2a < 0 \), the parabola is concave down. This simplifies to \( a < 0 \).

In summary, the concavity of a parabola is determined by the sign of the coefficient \( a \) in the quadratic function. If \( a > 0 \), the parabola is concave up, and if \( a < 0 \), it is concave down.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parabola

A parabola is a U-shaped curve that can open upwards or downwards, defined by a quadratic function of the form y = ax^2 + bx + c. The direction in which the parabola opens is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

Concavity

Concavity refers to the direction in which a curve bends. A function is concave up if it bends upwards like a cup, and concave down if it bends downwards like a cap. For a parabola, concavity is determined by the sign of the leading coefficient 'a' in the quadratic equation: positive 'a' indicates concave up, while negative 'a' indicates concave down.

Second Derivative Test

The second derivative test is used to determine the concavity of a function. For a quadratic function y = ax^2 + bx + c, the second derivative is a constant, 2a. If 2a > 0, the function is concave up, and if 2a < 0, it is concave down. This test helps in identifying the nature of the parabola's curvature.

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Textbook Question

Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.

Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

e. On what intervals (approximately) is f concave up?

Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

f. On what intervals (approximately) is f concave down?

Textbook Question

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

Textbook Question

Graph f(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2pi. Comment on the behavior of the graph of f in relation to the signs and values of f".

Textbook Question

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

Textbook Question

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

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114. Parabolasb. When is the parabola concave up? Concave down?

Key Concepts

Parabola

Concavity

Second Derivative Test

Watch next

114. Parabolas
b. When is the parabola concave up? Concave down?