Connecting to a Function, its first derivative, and its second derivative
Understanding Differentiation
- Differentiation provides a method to compute rates of change and determine the slope of a function at any point.
- The first derivative of a function, denoted as f'(x) or df/dx, represents the instantaneous rate of change at a point, or the slope of the tangent line to the curve at that point.
- A function will be increasing where its first derivative is positive, and decreasing where its first derivative is negative.
- Local maxima and minima of a function can be found where its first derivative equals zero.
Connecting a Function and Its First Derivative
- If f'(x) > 0, then the function f(x) is rising at x.
- If f'(x) = 0, then the function f(x) has a horizontal tangent at x.
- If f'(x) < 0, then the function f(x) is falling at x.
- Points of inflection are points at which the function changes concavity (i.e., switches between curving up and curving down). You can locate these by finding where the first derivative changes sign.
Understanding Second Derivative
- The second derivative, denoted as f''(x), provides information about the concavity or curvature of the graph.
- Specifically, if the second derivative is positive at a point, the function is concave up at that point. Conversely, if the second derivative is negative, the function is concave down.
- More intuitively, when a function is concave up, it curves upwards like the shape of a bowl. When a function is concave down, it curves downwards.
Connecting a Function, its First and Second Derivatives
- A function itself gives us information about the shape, position, and values of particular points.
- The first derivative gives us information about the rate of change of the function, identifying where the function has local extrema and inflection points.
- The second derivative confers information about the concavity or curvature of the function. This helps further classify local extrema and also identify inflection points.