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Flashcards: Determine Increasing/Decreasing Intervals

What is a critical value?

The point at which the derivative is equal to zero

A point of inflection

A local min or max

All of the above

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Flashcards: Determine Increasing/Decreasing Intervals

What is a critical value?

Your answer:

The point at which the derivative is equal to zero

Your answer:

A point of inflection

Your answer:

A local min or max

Correct answer:

All of the above

Explanation:

A point of inflection can be the point at which the derivative is equal to zero, a local minimum or maximum, and a point of inflection. A critical point will always be one that the tangent line to the original function is either completely horizontal or vertical. When the tangent line is completely horizontal, this will be a local maximum or minimum for the function. When the tangent line is completely vertical, then this will be a point of inflection. A point of inflection is a point in which there is a change of curvature in the graph of the function.

Flashcards: Determine Increasing/Decreasing Intervals

What is the first derivative test in regards to increasing/decreasing intervals?

If the critical value is positive, the function f(x) is increasing. If the critical value is negative, the function f(x) is decreasing.

You choose a number less than, and a number greater than the critical value. You plug these numbers into the derivative and if the solutions are positive, then the function f(x) is increasing but if the solutions are negative then the function f(x) is decreasing.

You choose a number less than the critical value. You plug this number into the derivative and if the solution is positive then the function f(x) is increasing, but if the solution is negative then the function f(x) is decreasing.

You choose a number greater than the critical value. You plug this number into the derivative and if the solution is positive then the function f(x) is increasing, but if the solution is negative then the function f(x) is decreasing.

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Flashcards: Determine Increasing/Decreasing Intervals

What is the first derivative test in regards to increasing/decreasing intervals?

Your answer:

If the critical value is positive, the function f(x) is increasing. If the critical value is negative, the function f(x) is decreasing.

Your answer:

Correct answer:

Explanation:

To find whether a function is decreasing or increasing along an interval, we look at the critical values and use what we call the first derivative test. Take the example f(x) = 2x^2+ 3x . The derivative would be f'(x) = 4x + 3 . To find the critical value we set the derivative equal to zero and solve for .

f'(x) = 4x + 3

0 = 4x + 3

-3 = 4x

$\frac{-3}{4} = x$

Now we have our critical point $\frac{-3}{4}$ . So we choose a number greater than this and plug it back into our derivative function.

f'(0) = 4(0) + 3

f'(0) = 3

The solution is positive. This means that the interval $(\frac{-3}{4}, \infty)$ on the graph of f(x)

f'(-1) = 4(-1) + 3

f'(-1) = -4 + 3 ,

f'(x) = -1

The solution is decreasing. This means that the interval $(-\infty, \frac{-3}{4})$ is decreasing on the graph of f(x) .

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Flashcards: Determine Increasing/Decreasing Intervals

Flashcards: Determine Increasing/Decreasing Intervals

Flashcards: Determine Increasing/Decreasing Intervals

Flashcards: Determine Increasing/Decreasing Intervals

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Flashcards: Determine Increasing/Decreasing Intervals

Flashcards: Determine Increasing/Decreasing Intervals

Flashcards: Determine Increasing/Decreasing Intervals

Flashcards: Determine Increasing/Decreasing Intervals

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