What does a tangent line lying below the curve indicate?

The function is concave up, and the tangent line approximation is an underestimate.

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What does a tangent line lying below the curve indicate?

The function is concave up, and the tangent line approximation is an underestimate.

What does a tangent line lying above the curve indicate?

The function is concave down, and the tangent line approximation is an overestimate.

How can you visually determine concavity from a graph?

If the graph curves upward, it's concave up. If it curves downward, it's concave down.

How does the slope of the tangent line relate to the derivative graph?

The slope of the tangent line at a point on the original function's graph is the y-value of the derivative graph at that point.

What does the graph of $f'(x)$ tell you about the slope of $f(x)$ ?

The y-value of $f'(x)$ at any point $x$ gives the slope of the tangent line to $f(x)$ at that point.

How does the concavity of $f(x)$ relate to the graph of $f''(x)$ ?

If $f''(x) > 0$ , $f(x)$ is concave up. If $f''(x) < 0$ , $f(x)$ is concave down.

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

$f'(x) = 0$ at that point.

How can you visually determine if a function is increasing or decreasing from its graph?

If the graph goes up from left to right, the function is increasing. If it goes down, it's decreasing.

How can you identify points of inflection on a graph?

Points where the concavity changes (from concave up to concave down, or vice versa).

What does the area under the curve of $f'(x)$ represent?

The change in the value of $f(x)$ .

What is the point-slope form of a line?

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

What is the linearization formula?

$L(x) = f(a) + f'(a)(x - a)$

How do you calculate the slope, m, for linearization?

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

How to find $f'(x)$ if given $f(x)$ ?

Differentiate $f(x)$ with respect to $x$ .

How to find the equation of tangent line?

Use $y - f(a) = f'(a)(x - a)$ , where $a$ is the x-coordinate of the point of tangency.

What is the formula to approximate $f(x)$ using linearization?

$f(x) \approx L(x) = f(a) + f'(a)(x-a)$

How do you find the derivative of a function at a specific point?

Evaluate $f'(x)$ at that point: $f'(a)$ .

What is the formula for the second derivative?

The derivative of the first derivative: $f''(x) = \frac{d}{dx} [f'(x)]$ .

How do you determine concavity using the second derivative?

If $f''(x) > 0$ , concave up; if $f''(x) < 0$ , concave down.

What is the formula for approximating f(a+h) using linearization?

$f(a+h) \approx f(a) + h*f'(a)$

Explain the concept of local linearity.

Zooming in on a curve until it appears as a straight line; using the tangent line to approximate function values near the point of tangency.

Explain how tangent lines are used for approximation.

The tangent line at a point closely approximates the function's values near that point. Plug in the x-value into tangent line equation to approximate the y-value.

How does the distance from the point of tangency affect accuracy?

Approximations are most accurate close to the point of tangency and become less accurate as you move further away.

Explain the relationship between concavity and over/underestimation.

Concave up means the tangent line is an underestimate. Concave down means the tangent line is an overestimate.

Describe the steps to approximate a function value using linearization.

Find the point of tangency, calculate the derivative at that point, build the tangent line equation, and plug in the x-value to approximate.

Explain why linearization works best for values close to the point of tangency.

The closer you are to the point of tangency, the more closely the tangent line resembles the original function.

Describe the relationship between the derivative and the tangent line.

The derivative of a function at a point is the slope of the tangent line to the function at that point.

How does the sign of the second derivative relate to concavity?

A positive second derivative indicates concave up, while a negative second derivative indicates concave down.

What does the first derivative tell you about the function?

The first derivative tells you about the increasing/decreasing behavior of the function.

Why is differentiability important for linearization?

Differentiability ensures the existence of a tangent line, which is essential for linearization.