Definitions
First Derivative
Second Derivative
Extrema
Linear Motion
Graphical Analysis
100
Possible minimum, maximum or point of inflection
Critical point
100
If f'(c) > 0, then f is...
Increasing at x = c.
100
If f''(c) > 0, then f(x) is...
Concave up at x = c
100
When is a critical point not an extremum?
If f' does not change sign at the critical point.
100
We say that v(t) is the first derivative of...
Position
100
Describe the first and second derivative at the point using the graph below.
f'(x) > 0
f"(x) < 0
200
Describe f at x=c if f'(c)>0 and f"(c)<0.
f is increasing and concave down.
200
f''(c) < 0 means f'(c) is...
Decreasing
200
Let f be a twice differentiable function. If f'(c) < 0 and f"(c) > 0, then f(x) is...
Concave up at x = c
200
Let f be a differentiable function. What are the steps for finding the absolute extremum on the interval [a,b]?
Find the critical points.
Test the critical points and the end points of f.
200
The second derivative of position.
a(t)
200
The following is the graph of the derivative of a differentiable function f. Describe the extrema based on this graph.
1 relative max and 1 relative min
300
A twice differentiable function is neither concave up or concave down at x = c if...
f"(x) = 0
300
If f'(x) = 0, and f'(a) < 0 and f'(b) > 0, for a<x<b, then x is a ______________ of f.
Relative minimum
300
True or false: It is always true that a point of inflection is where the second derivative is zero. Explain.
False. A point of inflection is where the second derivative changes sign.
300
If f'(c) = 0 and f"(c) < 0, then c is a...
Relative max
300
Fill in the blanks.
v(t) > 0 means a particle is __________.
v(t) < 0 means a particle is __________.
v(t) = 0 means the particle is __________.
Moving right or up
Moving left or down
At rest
300
The graph of the derivative of a differentiable function is given. What information can we get about the function based on this graph?
f is an increasing function and has two points of inflection
400
If f'(c) is undefined, but c is not a critical point, then...
c is not in the domain of f
400
The graph of h' is shown below. State the intervals where h is increasing, and where h is decreasing. Give a reason for your answer.
h is increasing on (A,B), (D,F), and (F,H)
because h' > 0
h is decreasing on (B,D) and (H,I) because h' < 0.
400
If f(c) is increasing and f(c) is concave down, then f''(c) is...
Negative
400
Let f be a differentiable function. If there is only one c such that f'(c) = 0 on [a,b], then c is...
An absolute extremum on [a,b]
400
Let x(t) be the position function of a particle. Describe the motion of the particle based on the information below:
x'(t) and x"(t) have different signs...
x'(t) and x"(t) have the same sign...
Slowing down when x'(t) and x"(t) have different signs.
Speeding up when x'(t) and x"(t) have the same sign.
400
The graph of a polynomial function f is given. Describe the graph of f', and state where the extrema and x-intercepts are located. Explain.
The graph is f' is a parabola opening down.
The maximum of the parabola is located at the point of inflection, b.
The x-intercepts are a and c, where the f has horizontal tangent lines.
500
State the MVT
If a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change over that interval.
500
Give three reasons why a function f is neither increasing or decreasing at x = c.
c is a critical point
c is not in the domain of f
f is not differentiable at c
500
What do we use the Second Derivative Test for? Give four reasons why it might be inconclusive.
The second derivative test is used to find extrema.
Inconclusive when:
f'(c) does not equal 0
f'(c) is undefined
f"(c)=0
f"(c) is undefined
500
Let f be a differentiable function. The graph of f' is given below. What are the absolute extrema on the interval [a,d]?
Absolute min at x = a
Absolute max at x = b
500
Let position of a particle x(t) be a differentiable function. The graph of x'(t) is shown below. On [a,e], state where the particle is speeding up, slowing down, and where it changes direction. Give a reason.
Speeding up: (a,b), (c,d), because x' and x'' have the same sign.
Slowing down: (b,c), (d,e), because x' and x'' have different signs.
Changes direction at x = c because x'(t) = 0 and x'(t) changes sign.
500
The graph of the derivative of a differentiable function is given. Describe the behavior of the graph of f.
f has a horizontal tangent line at x = c, but it is not an extremum. f is always decreasing.