Glossary
Approximation (using tangent line/linearization)
The process of estimating a function's value at a point by using the value of its tangent line at that same point.
Example:
We can use approximation with a tangent line to estimate √4.1 by finding the tangent line to f(x) = √x at x = 4.
Concave Down
A characteristic of a function's graph where its slope is decreasing, causing the curve to open downwards like a frown.
Example:
The graph of f(x) = -x² is concave down everywhere, so its tangent lines will always lie above the curve.
Concave Up
A characteristic of a function's graph where its slope is increasing, causing the curve to open upwards like a cup.
Example:
The graph of f(x) = x² is concave up everywhere, meaning its tangent lines will always lie below the curve.
Derivative
The instantaneous rate of change of a function with respect to its independent variable, representing the slope of the tangent line at any given point.
Example:
If f(x) = x³, its derivative f′(x) = 3x² tells us the slope of the tangent line at any x.
Linearization
The equation of the tangent line to a function at a specific point, used to approximate function values near that point.
Example:
The linearization L(x) = f(a) + f′(a)(x - a) provides a linear approximation of f(x) near x = a.
Local Linearity
The concept that a differentiable function, when viewed at a sufficiently small scale, appears to be a straight line.
Example:
When you zoom in on the graph of y = x² near x = 1, it looks almost perfectly straight, demonstrating local linearity.
Overestimate
An approximation that is greater than the true value of the function, typically occurring when the tangent line lies above the curve.
Example:
If a function is concave down, its tangent line will produce an overestimate for nearby function values.
Point-Slope Form
An algebraic form for the equation of a straight line, *y* - *y*₁ = *m*(*x* - *x*₁), where (*x*₁, *y*₁) is a point on the line and *m* is its slope.
Example:
To write the equation of a line with slope 3 passing through (2, 5), you'd use the point-slope form: y - 5 = 3(x - 2).
Tangent Line
A straight line that touches a curve at a single point and has the same slope as the curve at that point.
Example:
The line y = 2x - 1 is the tangent line to y = x² at x = 1.
Underestimate
An approximation that is less than the true value of the function, typically occurring when the tangent line lies below the curve.
Example:
For f(x) = x², the tangent line at x = 1 will give an underestimate for f(1.1) because the function is concave up.