Graphing Derivatives True or False Questions Flashcards
The rate at which one quantity changes with respect to another is the idea of a derivative.
True; this is the slope.
Every function has a derivative.
True; a function may not be locally linear at a point, but its derivative can still be created.
The notation dx/dy is commonly used for the derivative of an equation y=…
False; the correct notation is dy/dx.
The average velocity of a function is the ratio of the distance traveled and the time elapsed.
True
The average velocity of a function is the ratio of the “run” of a line to the “rise” of a line.
False; the average velocity of a function is the ratio of the “rise” of a line to the “run” of a line.
The average velocity over any interval is the slope of the secant line joining the endpoints of the interval.
True
Instantaneous velocity is the same as average velocity.
False; instantaneous velocity occurs at a point, while average velocity occurs over an interval.
Instantaneous velocity is measured by the slope of a tangent line to a curve at a particular point.
True
The term velocity and speed are the same.
False; velocity is a vector, so it has a direction assigned to its measure.
Speed is concerned with how fast an object is moving as well as the direction in which it is moving.
False; this is true of velocity.
The difference quotient,
f(a+h)-f(a)/h,
can be used to find the average rate of change of y with respect to x over the interval from a to a+h.
True
The instantaneous rate of change of a function f at point x=a is found by using the difference quotient,
f(a+h)-f(a)/h.
False; to find an instantaneous rate of change, the difference quotient needs a limit (as h->0) in front of it.
The derivative of a function is based on the idea of a secant line moving to become a tangent line.
True
To be differentiable at a point, a function should be “locally linear” at that point.
True
The vertex of an absolute value graph is an example of a point that is “locally linear.”
False; on an absolute value graph, at the vertex, the slope from the left does not equal the slope from the right. Therefore, it cannot be differentiable at the vertex.
The vertex of a parabola is an example of a point that is “locally linear.”
True
To determine the derivative of a function at a given point, you could draw a tangent line to the graph at that point and estimate its slope by counting squares on the graph.
True
The graph of a derivative of a function is increasing when the function is increasing.
False; the derivative of a function is positive when the function is increasing.
The graph of a derivative of a function has zero(s) at the value(s) of x where the function has inflection points.
False; zero(s) on the derivative corresponds to local extrema on the function (these could possibly involve an inflection point).
The graph of a second derivative of a function is increasing when the first derivative is concave up.
True
The graph of a second derivative of a function is negative when the graph of the first derivative is decreasing.
True
The graph of a derivative has maximums and minimums where the function has inflection points.
True
To compute a derivative numerically, you could use the symmetric difference quotient,
f(x)-2h/f(x+h).
False; the symmetric difference quotient is:
f(x+h)-f(x-h)/2h
The derivative of a function is a measure of velocity.
True
