Unit 3: The Derivative
Quotient Rule
(f/g)' = (f'g - g'f) / g^2
Product Rule
(fg)' = f'g + g'f
differentiable
A function is differentiable at a point if the derivative of the function exists at that point. At that point, the function must be continuous, and it must not have a "corner" ("teetering tangent") or a vertical tangent.
difference quotient
A quotient of differences, particularly the slope of a secant line: [f(x+h) - f(x)] / h
second derivative test
At a critical point where f' (x) = 0, if f'' (x) < 0, there is a relative max; if f'' (x) > 0, there is a relative min; if f'' (x) = 0, then the test fails.
instantaneous velocity
Instantaneous velocity is the velocity of an object at an instant in time. It is found by the slope of a position time graph at a point.
Rolle's Theorem
Suppose f is a function, differentiable for all points in the open interval (a, b) and continuous for all points in the closed interval [a, b]. If f(a) = f(b) = 0, then there is at least one number c between a and b for which f ' (c) = 0.
Mean Value Theorem
Suppose f is a function, differentiable for all points in the open interval (a, b), and continuous for all points in the closed interval [a, b]. Then there is at least one point c between a and b for which f'(c) = [f(b) - f(a)] / (b-a).
derivatives of sums and differences
The derivative of a sum is the sum of the derivatives. Also, the derivative of a difference is the difference of the derivatives.
local linearity
The tangent line to a curve lies close to the curve near the point of tangency--so close, in fact, that the curve and the line are almost the same. Very small parts of curves are nearly straight lines; the tangent line to a curve is nearly the same as the curve within a small neighborhood of the point of tangency.
differentiable function
a function that is differentiable at every point
chord
a line segment between two points on a curve
secant
a line segment between two points on a curve
tangent line
a line that touches a curve at one point and has the same slope as the curve at that point
inflection point
a point where the concavity changes; inflection points can only occur where f"(x) = 0 or where f"(x) does not exist
differential, or fractional, notation
dy/dx means "the derivative of y with respect to x"
derivative with respect to x
dy/dx, or the slope expressed as "rise over run"
prime notation
f '(x) or f ' means "the derivative of f(x)."
average rate of change
f(x) has an average rate of change over an interval [a, b] given by [f(b) - f(a)] / (b-a).
instantaneous rate of change
f(x) has an instantaneous rate of change at a given by lim x>a [f(x) - f(a)] / (x-a).
dot notation
notation in which x with a dot above it means "the derivative of x with respect to t"; this notation always means the derivative with respect to time; dot notation is traditional in physics settings
slope of a line
the amount of increase or decrease in the y (range) value for each increase or decrease in the x (domain) value; shown as the ratio of rise over run
concave up
the description of a curve when it is bowed or cupped in such a way that it would hold water
concave down
the description of a curve when it is bowed or cupped in such a way that it would spill water
concavity
the description of which way a curve is bowed or cupped
average velocity
the distance traveled divided by the time or travel
derived graph
the graph of the derivative of a function, often obtained by graphical differentiation
differentiating
the process of finding a derivative
derivative
the slope of a curve; the derivative of a function y = f(x) at a point (a, f(a)) is the slope of the tangent line to the curve y = f(x) at that point. The derivative of a function y = f(x) at a point (a, f(a)) is given by f'(a) = lim h>0 [f(x+h) - f(x)] / (x-a) The derivative for all values of x is a function in its own right and is given by dy/dx = f'(x) = lim h>0 [f(x+h) - f(x)] / h.
graphical differentiation
the use of a function's graph to sketch the graph of its derivative