Related Rates EVT and 1st and 2nd Derivative test

Related Rates, EVT and Derivatvies Tests

Multiple Choice

Who was the first "Major" Black filmmaker?

• Oscar Micheaux

• Spike Lee

• Maria P. Williams

• Melvin Van Peebles

Multiple Choice

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

• 3 ft/sec
• -7.2 ft/sec
• 7.2 ft/sec
• 12

Multiple Choice

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec.  How fast is the area of the outer circle changing when the diameter is 8 in?

• 80pi in/sec
• 20pi in/sec
• 60pi in/sec
• 40pi in/sec

Multiple Choice

Brandon is starting to clean up after a birthday party. He begins deflating each spherical balloon by puncturing a hole in each. The air leaves the balloon at a constant rate of 2 cm³/sec.  How long is the diameter when the radius is changing at a rate of 3 cm/sec?

• 0.424 cm

• 0.461 cm

• 1.333 cm
• 1.253 cm

Multiple Choice

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec.  How fast is the height of the water changing when the water is 2 ft deep?

• -9/(8pi) ft/sec
• 9/(8pi) ft/sec
• -8/(9pi) ft/sec
• 8/(9pi) f/tsec

Extreme Value Theorem

EVT

If a function 𝑓 is continuous over the interval [𝑎, 𝑏], then 𝑓 has at least one minimum value and at least one maximum value on [𝑎, 𝑏].

Absolute extrema use both critical points and end points x=a & x=b to find the absolute max and min by substituting into f(x)

​Absolute Extrema

f'(x) =0 and/or f'(x) DNE

Recall the 3 way something is
non differentiable.

Find Critical Points

EVT: How to use

Multiple Choice

Identify the critical points of the following function:
g(x)=2x³-3x²

• x=-1,1
• x=0,0
• x=0,-1
• x=0,1

Multiple Choice

When f'(x) changes from negative to positive, there is(are) ...

• a maximum.
• a minimum.
• no extrema.

Are found ​by analyzing your critical value and using a sign chart. This is called the 1st derivative test.

OR

Observing the graph of f(x) as seen here

Relative/Local Extrema

Means this point is a P.O.I aka point of inflection.

f'(x) =0 with no sign change

Find f'(x)=0 for critical points
f is Increasing when f'(x)>0 , Decreasing when f'(x)<0, Critical when f'(x)=0

Uses

1st derivate test

f''(x) =0 and has a sign change is where a point of inflection occurs.

What it means

f''(x) =0 and use a sign chart, similar to 1st derivative test

​
f''(x)>0
f''(x)<0

How to

2nd Derivatives

Recall
f''(x) is describing the curvature of the graph.

​

2nd Derivative

Multiple Choice

The concavity of a function is described by its _______________.

• first derivative
• second derivative
• third derivative
• expression

Multiple Select

If f'(x) = 0 what does that imply about the x value?

• It is a critical point, it is a possible max, min, or point of inflection.

• That the limit does not exist.

Multiple Choice

What will be true at an inflection point?  (select the best answer)

• f(x)=0
• f'(x)=0
• f''(x)=0
• The function is undefined

Multiple Choice

Over which interval is the graph of f(x) both decreasing and concave up?

• (B, C)

• (D, E)

• (C, D)

• (H, I)

Multiple Choice

Which of the following describes an interval of f(x) that is both decreasing and concave up?

• f '(x) < 0 and f "(x) < 0

• f '(x) < 0 and f "(x) > 0

• f '(x) > 0 and f "(x) > 0

• f '(x) > 0 and f "(x) < 0

Multiple Choice

For a function g(x), g''(3) = -8 indicates that g(x) is ____________ at x=3.

• increasing

• decreasing

• concave up

• concave down

At a critical point​ if f''(x) < 0, its concave down, which implies that at that point, f(x) has a local maximum

​At a critical point

At a critical point​ if f''(x) >0, its concave up, which implies that at that point, f(x) has a local minimum

At a critical point

2nd Derivative Test: Relative Extrema

Multiple Choice

when you check for sign changes at the critical points of a function, you are using the

• IVT

• MVT

• 1st Derivative Test

• 2nd Derivative Test

Multiple Choice

If (a,b) is a local maximum, then what will be true about f''(a)?

• It's positive
• It's negative
• It's zero
• Cannot be determined

Multiple Choice

Find the second derivative of the function:
f (x) =  2x - 5x⁶

• f ''(x)= 2 - 30x
• f ''(x) =  2-30x⁵
• f ''(x) = -30x⁵
• f ''(x) = -150x⁴

Multiple Choice

If the function f has a critical number at $$x=3$$   and $$f''\left(3\right)=5$$ , which of the following would be true?

• f has a local maximum at x=3

• f has a local minimum at x=3

• f has neither a local maximum or minimum at x=3

• there is not enough information to know if there is a local maximum or minimum at x=3

Multiple Choice

If the function f has a critical number at $$x=5$$   and $$f''\left(5\right)=-3$$ , which of the following would be true?

• f has a local maximum at x=5

• f has a local minimum at x=5

• f has neither a local maximum or minimum at x=5

• there is not enough information to know if there is a local maximum or minimum at x=5

Multiple Choice

How many extrema (maxes and mins) are in the picture?

• 2
• 3
• 4
• 5

Multiple Choice

Given the sign chart for f '(x), f(x) has ...

• a local maximum at x = -2.

• a local maximum at x = 4.

• local maxima at x = -2 and x = 4.

• no extrema.

Multiple Choice

Given the sign chart for f '(x), f(x) has ...

• a local maximum at x = -2.

• a local minimum at x = -2 and

  a local maximum at x = 4.

• a local maximum at x = -2 and

  a local minimum at x = 4.

• no extrema.

Multiple Choice

Given the graph of f '(x), f(x) has ...

• a local min at x = -6

• a local min at x = 2

• a local min at x = 2 and

  a local max at x = -6

• local mins at x = -2, 5 and

  a local max at x = 3

Multiple Choice

there is an absolute minimum at

• (4,5)

• (4,8)

• (-1,2)

• (2,-6)

Multiple Select

Find the absolute extrema of   $$f\left(x\right)=x^3-3x^2-24x+4$$  on the closed 

interval [ -4, 8].   Select ALL the correct answers.  

• Absolute minimum occurs when x = -4

• Absolute maximum occur when x = -2

• Absolute minimum occurs when x = 4

• Absolute maximum occurs when x = 4

• Absolute maximum occurs when x = 8

Multiple Choice

Who was the first black person to attended an all white school?

• Ruby Bridges
• Mary McLoud Bethune
• Linda Brown
• Sheyann Webb

Multiple Choice

Who invented the gas mask?

• Garrett Smith

• Garrett Monmouth

• Garrett Morgan

Multiple Choice

Who was the first black woman in space?

• Mae Jemison

• Shirley Chisholm

• Mary McLeoud Bethune

• Maxine Waters