What does the graph of $f'(x) > 0$ tell us about $f(x)$ ?

$f(x)$ is increasing.

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What does the graph of $f'(x) > 0$ tell us about $f(x)$ ?

$f(x)$ is increasing.

What does the graph of $f'(x) < 0$ tell us about $f(x)$ ?

$f(x)$ is decreasing.

What does the graph of $f'(x) = 0$ tell us about $f(x)$ ?

$f(x)$ has a horizontal tangent, which could be a local max, local min, or a stationary point.

What does the graph of $f''(x) > 0$ tell us about $f(x)$ ?

$f(x)$ is concave up.

What does the graph of $f''(x) < 0$ tell us about $f(x)$ ?

$f(x)$ is concave down.

What does the graph of $f(x) = c$ (a constant) tell us about $f'(x)$ ?

$f'(x) = 0$ (a horizontal line at y=0).

What does a sharp corner or cusp on the graph of $f(x)$ tell us about $f'(x)$ ?

$f'(x)$ does not exist at that point (non-differentiable).

If the graph of $f(x)$ is a straight line, what is the graph of $f'(x)$ ?

The graph of $f'(x)$ is a horizontal line, representing the constant slope of $f(x)$ .

What is the limit definition of the derivative?

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

What is the point-slope form of a line?

$y - y_1 = m(x - x_1)$

How do you find the slope (m) of a tangent line?

$m = f'(x_0)$ , where $x_0$ is the x-coordinate of the point of tangency.

How to find the equation of the tangent line?

Find $f'(x)$ . 2. Evaluate $f'(x_0)$ to find the slope $m$ . 3. Use point-slope form: $y - f(x_0) = m(x - x_0)$ .

Explain the relationship between a derivative and a tangent line.

The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point.

What does the derivative tell us about the behavior of a function?

The derivative indicates whether the function is increasing or decreasing, and the rate at which it is changing.

Explain the concept of a limit in the context of derivatives.

The limit allows us to find the instantaneous rate of change by considering what happens as the interval over which we calculate the average rate of change becomes infinitesimally small.

Why is the limit definition of the derivative important?

It provides the fundamental basis for understanding and calculating derivatives, and it connects the concept of slope to curves.

What is the significance of $h \to 0$ in the limit definition?

It means we are considering the slope of the secant line as the two points get infinitely close together, effectively becoming the tangent line.

Explain the concept of differentiability.

A function is differentiable at a point if its derivative exists at that point. This implies the function is continuous and has a well-defined tangent line.

What is the relationship between differentiability and continuity?

If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.

Explain how the derivative relates to real-world rates of change.

The derivative can model various real-world phenomena, such as velocity (rate of change of position), acceleration (rate of change of velocity), and rates of chemical reactions.

What does it mean for a tangent line to be horizontal?

A horizontal tangent line indicates that the derivative is zero at that point, implying a local maximum, local minimum, or a stationary point.

How do different notations of derivatives relate to each other?

$y'$ , $f'(x)$ , and $\frac{dy}{dx}$ all represent the same concept: the derivative of a function. They are used interchangeably depending on the context.

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What does the graph of $f'(x) > 0$ tell us about $f(x)$ ?

$f(x)$ is increasing.

What does the graph of $f'(x) < 0$ tell us about $f(x)$ ?

$f(x)$ is decreasing.

What does the graph of $f'(x) = 0$ tell us about $f(x)$ ?

$f(x)$ has a horizontal tangent, which could be a local max, local min, or a stationary point.

What does the graph of $f''(x) > 0$ tell us about $f(x)$ ?

$f(x)$ is concave up.

What does the graph of $f''(x) < 0$ tell us about $f(x)$ ?

$f(x)$ is concave down.

What does the graph of $f(x) = c$ (a constant) tell us about $f'(x)$ ?

$f'(x) = 0$ (a horizontal line at y=0).

What does a sharp corner or cusp on the graph of $f(x)$ tell us about $f'(x)$ ?

$f'(x)$ does not exist at that point (non-differentiable).

If the graph of $f(x)$ is a straight line, what is the graph of $f'(x)$ ?

The graph of $f'(x)$ is a horizontal line, representing the constant slope of $f(x)$ .

What is the limit definition of the derivative?

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

What is the point-slope form of a line?

$y - y_1 = m(x - x_1)$

How do you find the slope (m) of a tangent line?

$m = f'(x_0)$ , where $x_0$ is the x-coordinate of the point of tangency.

How to find the equation of the tangent line?

Find $f'(x)$ . 2. Evaluate $f'(x_0)$ to find the slope $m$ . 3. Use point-slope form: $y - f(x_0) = m(x - x_0)$ .

Explain the relationship between a derivative and a tangent line.

The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point.

What does the derivative tell us about the behavior of a function?

The derivative indicates whether the function is increasing or decreasing, and the rate at which it is changing.

Explain the concept of a limit in the context of derivatives.

The limit allows us to find the instantaneous rate of change by considering what happens as the interval over which we calculate the average rate of change becomes infinitesimally small.

Why is the limit definition of the derivative important?

It provides the fundamental basis for understanding and calculating derivatives, and it connects the concept of slope to curves.

What is the significance of $h \to 0$ in the limit definition?

It means we are considering the slope of the secant line as the two points get infinitely close together, effectively becoming the tangent line.

Explain the concept of differentiability.

A function is differentiable at a point if its derivative exists at that point. This implies the function is continuous and has a well-defined tangent line.

What is the relationship between differentiability and continuity?

If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.

Explain how the derivative relates to real-world rates of change.

The derivative can model various real-world phenomena, such as velocity (rate of change of position), acceleration (rate of change of velocity), and rates of chemical reactions.

What does it mean for a tangent line to be horizontal?

A horizontal tangent line indicates that the derivative is zero at that point, implying a local maximum, local minimum, or a stationary point.

How do different notations of derivatives relate to each other?

$y'$ , $f'(x)$ , and $\frac{dy}{dx}$ all represent the same concept: the derivative of a function. They are used interchangeably depending on the context.