Graphing Derivatives True or False Questions

Question 1

The rate at which one quantity changes with respect to another is the idea of a derivative.

Answer 1

True; this is the slope.

Question 2

Every function has a derivative.

Answer 2

True; a function may not be locally linear at a point, but its derivative can still be created.

Question 3

The notation dx/dy is commonly used for the derivative of an equation y=…

Answer 3

False; the correct notation is dy/dx.

Question 4

The average velocity of a function is the ratio of the distance traveled and the time elapsed.

Answer 4

True

Question 5

The average velocity of a function is the ratio of the “run” of a line to the “rise” of a line.

Answer 5

False; the average velocity of a function is the ratio of the “rise” of a line to the “run” of a line.

Question 6

The average velocity over any interval is the slope of the secant line joining the endpoints of the interval.

Answer 6

True

Question 7

Instantaneous velocity is the same as average velocity.

Answer 7

False; instantaneous velocity occurs at a point, while average velocity occurs over an interval.

Question 8

Instantaneous velocity is measured by the slope of a tangent line to a curve at a particular point.

Answer 8

True

Question 9

The term velocity and speed are the same.

Answer 9

False; velocity is a vector, so it has a direction assigned to its measure.

Question 10

Speed is concerned with how fast an object is moving as well as the direction in which it is moving.

Answer 10

False; this is true of velocity.

Question 11

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.

Answer 11

True

Question 12

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.

Answer 12

False; to find an instantaneous rate of change, the difference quotient needs a limit (as h->0) in front of it.

Question 13

The derivative of a function is based on the idea of a secant line moving to become a tangent line.

Answer 13

True

Question 14

To be differentiable at a point, a function should be “locally linear” at that point.

Answer 14

True

Question 15

The vertex of an absolute value graph is an example of a point that is “locally linear.”

Answer 15

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.

Question 16

The vertex of a parabola is an example of a point that is “locally linear.”

Answer 16

True

Question 17

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.

Answer 17

True

Question 18

The graph of a derivative of a function is increasing when the function is increasing.

Answer 18

False; the derivative of a function is positive when the function is increasing.

Question 19

The graph of a derivative of a function has zero(s) at the value(s) of x where the function has inflection points.

Answer 19

False; zero(s) on the derivative corresponds to local extrema on the function (these could possibly involve an inflection point).

Question 20

The graph of a second derivative of a function is increasing when the first derivative is concave up.

Answer 20

True

Question 21

The graph of a second derivative of a function is negative when the graph of the first derivative is decreasing.

Answer 21

True

Question 22

The graph of a derivative has maximums and minimums where the function has inflection points.

Answer 22

True

Question 23

To compute a derivative numerically, you could use the symmetric difference quotient,
f(x)-2h/f(x+h).

Answer 23

False; the symmetric difference quotient is:
f(x+h)-f(x-h)/2h

Question 24

The derivative of a function is a measure of velocity.

Answer 24

True

Question 25

The derivative of a function is a measure of acceleration.

Answer 25

False; the second derivative of a function is a measure of acceleration.

Question 26

Critical numbers are locations where the derivative of a function is positive.

Answer 26

False; critical numbers occur when f’(x)=0 or f’(x)=DNE.

Question 27

An inflection point is a zero on the graph of the first derivative.

Answer 27

False; it is a maximum, minimum, or a shelf (local extrema) on the graph of the first derivative.

Question 28

The derivative of the sine function is the cosine function.

Answer 28

True

Question 29

The derivative of the cosine function is the sine function.

Answer 29

False; the derivative of the cosine function is -sinx.