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What does an increasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave up.

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What does an increasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave up.

What does a decreasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave down.

If the graph of $f''(x)$ is above the x-axis, what does this say about the concavity of $f(x)$ ?

$f(x)$ is concave up.

If the graph of $f''(x)$ is below the x-axis, what does this say about the concavity of $f(x)$ ?

$f(x)$ is concave down.

How can you identify a possible inflection point on the graph of $f''(x)$ ?

Look for points where the graph crosses the x-axis (i.e., where $f''(x) = 0$ ).

How can you identify intervals of concave up on the graph of $f'(x)$ ?

Look for intervals where the slope of $f'(x)$ is positive.

How can you identify intervals of concave down on the graph of $f'(x)$ ?

Look for intervals where the slope of $f'(x)$ is negative.

What does a horizontal tangent on the graph of $f'(x)$ indicate?

A possible inflection point on the graph of $f(x)$ .

If $f'(x)$ is a straight line with a positive slope, what does this indicate about $f(x)$ ?

$f(x)$ is concave up and has a constant rate of change of its slope.

If $f'(x)$ is a straight line with a negative slope, what does this indicate about $f(x)$ ?

$f(x)$ is concave down and has a constant rate of change of its slope.

What condition on $f''(x)$ indicates concave up?

$f''(x) > 0$

What condition on $f''(x)$ indicates concave down?

$f''(x) < 0$

What equation is used to find possible inflection points?

$f''(x) = 0$

How to find the first derivative?

Apply differentiation rules (e.g., power rule, product rule, chain rule) to $f(x)$ to find $f'(x)$

How to find the second derivative?

Differentiate the first derivative: $f''(x) = (f'(x))'$

What is the power rule?

$\frac{d}{dx} x^n = nx^{n-1}$

What is the product rule?

$\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

What is the chain rule?

$\frac{d}{dx} [f(g(x))] = f'(g(x)) cdot g'(x)$

What is the derivative of a constant?

$\frac{d}{dx} c = 0$

What is the derivative of $e^x$ ?

$\frac{d}{dx} e^x = e^x$

Define concavity.

The direction a curve opens; concave up faces upward, concave down faces downward.

What is a point of inflection?

A point where a function changes concavity.

Define concave up in terms of the first derivative.

The slopes of tangent lines are increasing, or $f'(x)$ is increasing.

Define concave down in terms of the first derivative.

The slopes of tangent lines are decreasing, or $f'(x)$ is decreasing.

How is concavity related to the second derivative?

Concave up: $f''(x) > 0$ . Concave down: $f''(x) < 0$ .

What is a possible point of inflection?

A point where $f''(x) = 0$ .

What must be true at a true point of inflection?

$f$ must change concavity and $f''(x) = 0$ .

What does a positive second derivative indicate?

The function is concave up.

What does a negative second derivative indicate?

The function is concave down.

How to find possible inflection points?

Set the second derivative equal to zero and solve for x: $f''(x)=0$ .

All Flashcards

What does an increasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave up.

What does a decreasing $f'(x)$ graph indicate about $f(x)$ ?

$f(x)$ is concave down.

If the graph of $f''(x)$ is above the x-axis, what does this say about the concavity of $f(x)$ ?

$f(x)$ is concave up.

If the graph of $f''(x)$ is below the x-axis, what does this say about the concavity of $f(x)$ ?

$f(x)$ is concave down.

How can you identify a possible inflection point on the graph of $f''(x)$ ?

Look for points where the graph crosses the x-axis (i.e., where $f''(x) = 0$ ).

How can you identify intervals of concave up on the graph of $f'(x)$ ?

Look for intervals where the slope of $f'(x)$ is positive.

How can you identify intervals of concave down on the graph of $f'(x)$ ?

Look for intervals where the slope of $f'(x)$ is negative.

What does a horizontal tangent on the graph of $f'(x)$ indicate?

A possible inflection point on the graph of $f(x)$ .

If $f'(x)$ is a straight line with a positive slope, what does this indicate about $f(x)$ ?

$f(x)$ is concave up and has a constant rate of change of its slope.

If $f'(x)$ is a straight line with a negative slope, what does this indicate about $f(x)$ ?

$f(x)$ is concave down and has a constant rate of change of its slope.

What condition on $f''(x)$ indicates concave up?

$f''(x) > 0$

What condition on $f''(x)$ indicates concave down?

$f''(x) < 0$

What equation is used to find possible inflection points?

$f''(x) = 0$

How to find the first derivative?

Apply differentiation rules (e.g., power rule, product rule, chain rule) to $f(x)$ to find $f'(x)$

How to find the second derivative?

Differentiate the first derivative: $f''(x) = (f'(x))'$

What is the power rule?

$\frac{d}{dx} x^n = nx^{n-1}$

What is the product rule?

$\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

What is the chain rule?

$\frac{d}{dx} [f(g(x))] = f'(g(x)) cdot g'(x)$

What is the derivative of a constant?

$\frac{d}{dx} c = 0$

What is the derivative of $e^x$ ?

$\frac{d}{dx} e^x = e^x$

Define concavity.

The direction a curve opens; concave up faces upward, concave down faces downward.

What is a point of inflection?

A point where a function changes concavity.

Define concave up in terms of the first derivative.

The slopes of tangent lines are increasing, or $f'(x)$ is increasing.

Define concave down in terms of the first derivative.

The slopes of tangent lines are decreasing, or $f'(x)$ is decreasing.

How is concavity related to the second derivative?

Concave up: $f''(x) > 0$ . Concave down: $f''(x) < 0$ .

What is a possible point of inflection?

A point where $f''(x) = 0$ .

What must be true at a true point of inflection?

$f$ must change concavity and $f''(x) = 0$ .

What does a positive second derivative indicate?

The function is concave up.

What does a negative second derivative indicate?

The function is concave down.

How to find possible inflection points?

Set the second derivative equal to zero and solve for x: $f''(x)=0$ .