By the end of the unit the student will learn that...
The difference quotients (f(a+h)-f(a))/h or (f(x)-f(a))/(x-a) express the average rate of change of a function over an interval.
The instantaneous rate of change of a function at a point can be expressed by lim [(f(a+h)-f(a))/h] or lim[(f(x)-f(a))/(x-a)], provided that the limit exists. These are common forms of the definition of the derivative and are denoted f’(a).
The derivative of f is the function whose value at x is (f(a+h)-f(a))/h provided that this limit exists.
For y = f(x), notations for the derivative include dx/dy, f’(x), and y’.
The derivative can be represented graphically, numerically, analytically, and verbally.
The derivative at a point can be estimated from information given in tables or graphs.
Direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, and cosine.
Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, and trigonometric.
Sums, differences, products, and quotients of functions can be differentiated using derivative rules.
The chain rule provides a way to differentiate composite functions.
Differentiating f’ produces the second derivative f’’, provided the derivative of f’ exists; repeating this process produces higher order derivatives of f.
Higher order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include d²y/dx², f’’(x), and y’’. Higher order derivatives can be denoted dny/dxn,or f(n)(x).
Key features of the graphs of f, f ’, and f ’’ are related to one another.
A continuous function may fail to be differentiable at a point in its domain.
If a function is differentiable at a point, then it is continuous at that point.
The unit for f’(x) is the unit for f divided by the unit for x.
The derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable.
The derivative at a point is the slope of the line tangent to a graph at that point on the graph.
The tangent line is the graph of a locally linear approximation of the function near the point of tangency.
The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration.
The derivative can be used to solve related rate problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known.
The derivative can be used to express information about rates of change in applied contexts.