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.

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.

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

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

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

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

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

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

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

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

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

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