Approximating a function with a tangent line near a specific point.

The equation of the tangent line used to approximate function values.

The point where the tangent line touches the curve of the function.

The direction of the curve of a function (upward or downward).

When the tangent line approximation is greater than the actual function value.

When the tangent line approximation is less than the actual function value.

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

A line that touches a curve at a single point and has the same slope as the curve at that point.

The rate of change of the function at the point of tangency, equal to the derivative at that point.

The function has a derivative at every point in its domain; it is smooth and continuous.

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

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

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

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.

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

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

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

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

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

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

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 $\frac{f(b)-f(a)}{b-a}$.