Figure 1 shows the graph of a function



Drawing Functions based upon Derivative Sign Information

Table 1: 4 Basic Curve Shapes and the attendant Derivative Signs

| |[pic] |[pic] |

|[pic] | | |

| |[pic] |[pic] |

|[pic] | | |

| |[pic] |[pic] |

Example

Sketch onto Figure 1 a continuous curve, [pic], that has the following properties.

• [pic]

• [pic] over [pic] and [pic]

• [pic] over [pic] and [pic]

• [pic] over [pic] and [pic]

• [pic] over [pic]

Four different functions are shown in figures 3 - 6. In each question on this page a property is stated that is true for only one of the functions shown in figures 3 - 6. For each property, state the figure number that shows the function with the stated property. Write below each question a brief explanation of how you made your determination.

a. The function whose first derivative is increasing over the interval [pic] is shown in Figure

b. The function whose first derivative is negative over the interval [pic] is shown in Figure

c. The function whose antiderivative is decreasing over [pic] is shown in Figure

d. The function whose first derivative has a local minimum point at [pic] is in Figure

e. The function whose antiderivative is concave up over [pic] is shown in Figure

f. The function whose antiderivative has a local maximum point at [pic] is in Figure

The graph of a first derivative, [pic], is shown in Figure 21. Sketch onto Figure 22 the function [pic] given that [pic] and [pic].

Figure 2 shows the graph of the second derivative of a function named g. Answer each of the following questions. In each case, explain your reasoning.

• Where is g concave up?

• Where does g have its point(s) of inflection?

• Rank the four numbers [pic] in increasing order.

• Suppose that [pic]. Is g increasing or decreasing at the point where [pic]?

Figure 5 is the graph of a function named f. F is an antiderivative of f. Answer each of the following questions about F.

• On what intervals is F increasing? Explain.

• At what values of x does F have a local maximum or local minimum? Explain.

• On what intervals is F concave up? concave down? Explain.

• At what values of x does F have an inflection point? Explain.

Water flows at a constant rate into a large conical tank (pointy end down (). Let [pic] (in [pic] ) and [pic] (in ft) be, respectively, the volume of water in the tank and the height of the water in the tank t minutes after the water begins to flow. Suppose that five minutes after the water begins to flow the tank is one quarter full. For each of the following expressions, state the units on the expression and state whether the value is positive, negative, or zero. Explain!

[pic] [pic] [pic] [pic]

Figures A-F show 6 different functions. The first derivative of one of these functions is [pic]. Which one is a graph of [pic]? No work need be shown.

Each of the following sentences is true if one of the words/phrases in Table 1 is inserted into the blank. Find the proper word/phrase for each of the blanks. Read each sentence carefully!!

• If[pic] is negative at every value of x on [pic], then [pic] is

over the entire interval[pic].

• If [pic] is positive at every value of x over [pic], then [pic] is over the entire interval [pic].

• If [pic] is increasing over the entire interval [pic], then [pic] is over the entire interval [pic].

• If the slope of [pic] is increasing over the entire interval [pic], then [pic] is

over the entire interval [pic].

• If the slope of [pic] is positive over the entire interval [pic], then [pic] is

over the entire interval [pic].

• If [pic] is continuous and has a local minimum at the point where [pic],

then [pic] is immediately to the right of [pic].

• If the slope of [pic] is increasing over the entire interval [pic], then at any point along the

interval [pic], [pic] is .

(This question is continued on page 7.)

• If [pic] is differentiable and has a local maximum point at [pic] then the tangent line to

[pic] at the point where [pic] is .

• If [pic] is decreasing over the entire interval [pic] then [pic] is .

• If [pic] is concave up over the entire interval [pic], then [pic] is over the entire interval [pic].

• If the slope of [pic] is negative over the entire interval [pic],

then [pic] is .

• If [pic] is concave up over the entire interval [pic],

then [pic] is .

• If [pic] has a local maximum at the point where [pic], then the slope of the tangent line to

[pic] is at [pic].

Suppose that [pic] and [pic] for all values of t. Over what interval(s) is the function [pic] decreasing? Explain!

Sketch a curve that satisfies each of the indicated properties. (Sketch a different curve for each stated property. ()

The fuel consumption rate (measured in gallons per hour) of a car traveling at a speed of v mph is given by the function [pic].

a. What is the unit on [pic]?

b. What is the unit on [pic]?

c. What is the practical meaning of [pic]?

d. What is the practical meaning of [pic]?

d. What is the practical meaning of [pic]?

-----------------------

x

Figure 4

y

x

Figure E

[pic] is decreasing everywhere yet [pic] is nondifferentiable at [pic]

Figure 14:

y

x

[pic] is increasing everywhere yet [pic]

Figure 3

Figure 22

y

x

Figure D

y

x

Figure C

y

[pic]

[pic]

x

Figure F

y

x

Figure 3

Table 1: Blank Options

|Increasing |Decreasing |

|Concave Up |Concave Down |

|Positive |Negative |

|Zero |"Zero or UD" |

| |(This is one choice) |

|"Positive, Zero or UD" |"Negative, Zero or UD" |

|(This is one choice) |(This is one choice) |

|Horizontal |Vertical |

Note: UD is short for undefined

x

y

Figure 15:

[pic] has a local minimum at [pic] but [pic]

x

y

Figure 13:

Figure 21

y

x

Figure B

y

x

Figure A

y

Table 1: Blank Options

|Increasing |Decreasing |

|Concave Up |Concave Down |

|Positive |Negative |

|Zero |"Zero or UD" |

| |(This is one choice) |

|"Positive, Zero or UD" |"Negative, Zero or UD" |

|(This is one choice) |(This is one choice) |

|Horizontal |Vertical |

Note: UD is short for undefined

Figure 1: [pic]

y

x

Figure 6

Figure 5

Figure 4

Figure 5

Figure 6

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