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Unit 3 - Applications of Differentiation
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Terms in this set (30)
First Derivative Test
a method for determining whether a critical point is a minimum, maximum, or neither
Absolute Maximum
the highest point over the entire domain of a function or relation; The first derivative test and the second derivative test are common methods used to find maximum values of a function. Testing endpoints is needed for closed intervals.
Absolute Minimum
the lowest point over the entire domain of a function or relation; The first derivative test and the second derivative test are common methods used to find maximum values of a function. Testing endpoints is needed for closed intervals.
Acceleration
the rate of change of velocity over time; For motion along the number line, acceleration is a scalar. For motion on a plane or through space, acceleration is a vector. Acceleration is the derivative of velocity with respect to time.
Concavity
refers to the directionality of a curve; Note: If a curve is concave up (convex), the graph of the curve is bent upward, like an upright bowl. If a curve is concave down (or simply concave), then the graph of the curve is bent down, like a bridge.
Critical Number/Critical Point/Critical Value
a point (x, y) on the graph of a function at which the derivative is either 0 or undefined; can be a minimum or maximum, but it may be neither
Decreasing Function
a function with a graph that moves downward as it is followed from left to right; e.g., any line with a negative slope is decreasing; Note: If a function is differentiable, then it is decreasing at all points where its derivative is negative.
Extreme Value Theorem
a theorem which guarantees the existence of an absolute max and an absolute min for any continuous function over a closed interval
Extremum
an extreme value of a function; the minima and maxima of a function; may be either relative (local) or absolute (global); Note: The first derivative test and the second derivative test are common methods used to find extrema.
Absolute Maximum
the highest point over the entire domain of a function or relation; Note: The first derivative test and the second derivative test are common methods used to find maximum values of a function.
Absolute Minimum
the lowest point over the entire domain of a function or relation; Note: The first derivative test and the second derivative test are common methods used to find minimum values of a function.
Increasing Function
a function with a graph that goes up as it is followed from left to right; any line with a positive slope is increasing; Note: If a function is differentiable, then it is increasing at all points where its derivative is positive.
Indeterminate Expression
an undefined expression which can have a value if arrived at as a limit
Inflection Point
a point at which a curve changes from concave up to concave down, or vice-versa; Note: If a function has a second derivative, the value of the second derivative is either 0 or undefined at each of that function's inflection points.
Instantaneous Acceleration
the rate at which an object's instantaneous velocity is changing at a particular moment; found by taking the derivative of the velocity function; Note: For motion on the number line, instantaneous acceleration is a scalar. For motion on a plane or in space, it is a vector."
Instantaneous Velocity
the rate at which an object is moving at a particular moment; same as the derivative of the function describing the position of the object at a particular time; Note: For motion on the number line, instantaneous velocity is a scalar. For motion on a plane or in space, it is a vector.
L'Hospital's Rule
a technique used to evaluate limits of fractions that evaluate to the indeterminate expressions 0/0 and positive/negative infinity over infinity; done by finding the limit of the derivatives of the numerator and denominator
Local Behavior
the appearance or properties of a function, graph, or geometric figure in the immediate neighborhood of a particular point; usually this refers to any appearance or property that becomes more apparent as you zoom in on the point; For example, as you zoom in to the graph of y = x^2 at any point, the graph looks more and more like a line. Thus, we say that y = x^2 is locally linear. We say this even though the graph is not actually a straight line.
Relative Maximum
the highest point in a particular section of a graph; also called local maximum; Note: The first derivative test and the second derivative test are common methods used to find maximum values of a function.
Relative Minimum
the lowest point in a particular section of a graph; also called local minimum; Note: The first derivative test and the second derivative test are common methods used to find maximum values of a function.
Maximize
to find the largest possible value
Maximum of a Function
either a relative (local) maximum or an absolute (global) maximum
Mean Value Theorem
a theorem of calculus that relates values of a function to a value of its derivative; essentially the theorem states that for a coninunous, differentiable function, there is a tangent line parallel to any secant line
Minimize
to find the smallest possible value
Minimum of a Function
either a relative (local) minimum or an absolute (global) minimum
Related Rates
a class of problems in which rates of change are related by means of differentiation
Second Derivative
the derivative of a derivative
Second Derivative Test
a method for determining whether a critical point is a relative minimum or maximum
Speed
distance covered per unit of time; a nonnegative scalar; Note: For motion in one dimension, such as on a number line, speed is the absolute value of velocity. For motion in two or three dimensions, speed is the magnitude of the velocity vector.
Velocity
the rate of change of the position of an object; Note: For motion in one dimension, such as along the number line, velocity is a scalar. For motion in two dimensions or through three-dimensional space, velocity is a vector.
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