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.

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.

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

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

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

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

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

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

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

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

1. Find the derivative of the function. 2. Evaluate the derivative at the given x-value to find the slope. 3. Use the point-slope form to write the equation of the line.

1. Find the tangent line at a nearby point $a$. 2. Plug $x$ into the tangent line equation to find the approximate value.

1. Find the second derivative of the function. 2. Determine the concavity at the point of tangency. 3. Concave up = underestimate, concave down = overestimate.

1. Substitute $x = 1.1$ into the equation of the tangent line. 2. Solve for y. The y-value is the approximate value of $f(1.1)$.

1. Find two points near $x=3$ in the table. 2. Calculate the slope between those two points using the difference quotient.

1. Find $f(a)$. 2. Find $f'(x)$. 3. Find $f'(a)$. 4. Use the point-slope form: $y - f(a) = f'(a)(x - a)$.

1. Find the tangent line at $x=a$: $y - f(a) = f'(a)(x - a)$. 2. Substitute $x = a+h$ into the tangent line equation. 3. Solve for $y$.

1. Find the second derivative, $f''(x)$. 2. Determine the sign of $f''(x)$. 3. If $f''(x) > 0$, concave up. If $f''(x) < 0$, concave down.

1. Find the derivative, $f'(x)$. 2. Evaluate the derivative at $x=a$: $f'(a)$.

1. Find the equation of the tangent line. 2. Substitute the x-value into the equation of the tangent line. 3. Solve for y.

Point-slope: $y - y_1 = m(x - x_1)$ | Linearization: $L(x) = f(a) + f'(a)(x - a)$. Both achieve the same result, but point-slope is often easier to remember.

Overestimate: Tangent line is above the curve (concave down). | Underestimate: Tangent line is below the curve (concave up).

First Derivative: Rate of change of the function, slope of the tangent line. | Second Derivative: Rate of change of the first derivative, concavity of the function.

Concave Up: Curve opens upwards, $f''(x) > 0$, underestimate. | Concave Down: Curve opens downwards, $f''(x) < 0$, overestimate.

Function: The original curve. | Tangent Line: A linear approximation of the function at a specific point.

Differentiability: Function has a derivative at every point. | Continuity: Function has no breaks or jumps.

$f(x)$: The original function. | $f'(x)$: The derivative of the function, representing the rate of change.

Approximation: An estimate using linearization. | Actual Value: The exact value of the function.

Slope: The steepness of the tangent line, determined by $f'(x)$. | Concavity: The direction of the curve, determined by $f''(x)$.

Tangent Line: Touches the curve at one point, slope is $f'(a)$. | Secant Line: Touches the curve at two points, slope is $\frac{f(b)-f(a)}{b-a}$.