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
The derivative of a function is a measure of acceleration.
False; the second derivative of a function is a measure of acceleration.
Critical numbers are locations where the derivative of a function is positive.
False; critical numbers occur when f’(x)=0 or f’(x)=DNE.
An inflection point is a zero on the graph of the first derivative.
False; it is a maximum, minimum, or a shelf (local extrema) on the graph of the first derivative.
The derivative of the sine function is the cosine function.
True
The derivative of the cosine function is the sine function.
False; the derivative of the cosine function is -sinx.
