7.3.1 The Connection Between Slope and Optimization

Question 1

The Connection Between Slope and Optimization

Answer 1

• On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing.
• Values that make the derivative of a function equal to 0 are candidates for the locations of maxima and minima of the function.

Question 2

note 1

Answer 2

• The behavior of the tangent line gives insight into the
  behavior of a curve.
• Notice that for an interval in which the curve is increasing, the tangent lines must be positively sloped. Likewise, when the curve is decreasing, the tangent lines must be negatively sloped.
• In addition, when a curve changes from increasing to
  decreasing or from decreasing to increasing, the curve can attain a maximum or minimum value. These points occur where the derivative is equal to 0 (or undefined).
• Here is an example of this concept in action.
• Notice that the derivative is negative for x = 0, but positive for x = 10.
• Not only does this give you insight to the shape of the curve, it also indicates that there is probably a minimum point between these two x-values.

Question 3

note 2

Answer 3

• One application of this process is called optimization. If you can find the maximum or minimum value of a function, then you can determine where a function will produce the greatest profit or the least amount of waste.
• Start by finding the derivative of the function.
• Next, set the derivative equal to 0 and solve for x.
• Once you know the candidates for the maximum or minimum points, you can test the derivative on either side of those candidates to see if the function is increasing or decreasing on that side.
• If a function is increasing to the left and decreasing to the right of a candidate, then there is a maximum value at that point.
• If a function is decreasing to the left and increasing to the right of a candidate, then there is a minimum value at that point.
• It is a good idea to make some sort of chart for this
  information so you will remember where the function
  increases and decreases.
• Once you know where the maximum value is located, you can plug that value into the function to find the actual maximum value.

Question 4

Is the function f(x)=cos^2xsin3x increasing,decreasing, or neither at the point x=π?

Answer 4

decreasing

Question 5

Is this function increasing, decreasing, or neither at x = −1?

Answer 5

decreasing

Question 6

Suppose you are given that f (−2) = −10, f (2) = 5, f ′(−2) = 3, and f ′(2) = −2. Is f (x) increasing, decreasing, or neither at the point x = −2?

Answer 6

increasing

Question 7

A certain jet plane consumes fuel at a rate (in lbs / hour) given by the function
F (x) = x ^4 − 20x^ 3 + 28,000, where x is the amount of fuel additive used in ounces.

What is the minimum rate of fuel consumption of the plane (in lbs / hour)?

Answer 7

11,125

Question 8

Is the point (0, 4) a minimum point of the function? Why or why not?
y=x^3+4x/x

Answer 8

No, because the function is not defined at the point (0, 4).

Question 9

Consider the function y=∣∣3√x∣∣.Is (0, 0) a minimum point? Why or why not?

Answer 9

Yes, since the function is defined at the point, decreases to the left, and increases to the right.

Question 10

Consider this graph of f (x).

What can you say about the value of the derivative at the function’s absolute minimum point?

Answer 10

f ′(x) = 0

Question 11

The area of a rectangle is given by the function A (x) = 8x − x ^2, where x is the length of one of the sides. What is the maximum area of this rectangle?

Answer 11

16

Question 12

Suppose that the costs for a small trucking company are given by the function C(d)=30+2d+72/d, where d is the number of trucks used. How many trucks should the company use in order to minimize costs?

Answer 12

6 trucks