[Notes & HW Answers] 6.1 Antiderivatives Graphically and Numerically

[Notes & HW Answers] 6.1 Antiderivatives Graphically and Numerically.pdf

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[Prepwork 6.1]

Q1. Enter two different antiderivatives of 𝑓=2𝑥.

A: Antiderivative 1 = x^2+1

Antiderivative 2 = x^2+2

Q2. Suppose that the graph of 𝑓′f′ is given below.

 

At what 𝑥 value does 𝑓 cease being linear? 𝑥= 1
At what 𝑥 value does the maximum value of 𝑓 occur? 𝑥= 3 

Q3. Suppose that 𝐹′(𝑡)=𝑡cos(𝑡) and 𝐹(0)=3. Use the data and method from example 5 in the text to estimate each of the following.

𝐹(0.4)= 3.077
𝐹(0.5)= 3.117 

[HW 6.1]

Q1. Consider the graph of 𝑔(𝑡) shown below.

If the areas A, B, and C are all defined to be the areas between the graph of 𝑔(𝑡) and the 𝑥x-axis and are, respectively, 5, 19, and 41, carefully sketch the graph of an antiderivative 𝐺(𝑡) of 𝑔(𝑡) satisfying 𝐺(0)=15. Label each critical point of 𝐺(𝑡) with its coordinates.

Then enter the critical points below, as a list of ordered pairs (that is, if you found critical points at 𝑥=2, 𝑦=3 and 𝑥=3, 𝑦=4, you would enter (2,3), (3,4)).
critical points = 

A: f'(x) = (0,15),(1,20),(3,1),(5,42)

 

Q2. Given the values of the derivative 𝑓′(𝑥) in the table and that 𝑓(0)=100, find or estimate 𝑓(𝑥) for 𝑥=0,2,4,6. 

𝑥	0	2	4	6
𝑓′(𝑥)	9	22	31	35
𝑓(𝑥)	100	144	206	276

Q3. For the graph of 𝑓(𝑥) shown below, sketch two functions 𝐹 with 𝐹′(𝑥)=𝑓(𝑥). In one let 𝐹(0)=0; in the other, let 𝐹(0)=1. Mark 𝑥1, 𝑥2 and 𝑥3 on the 𝑥-axis of your graph. Identify local maxima, minima and inflection points of 𝐹(𝑥).

(a) At which point does 𝐹(𝑥) achieve its largest value?
A. 𝑥1
B. 𝑥2
C. 𝑥3

(b) At which point does 𝐹(𝑥)F(x) achieve its smallest value?
A. 𝑥1
B. 𝑥2
C. 𝑥3

For the following questions, consider only the interior points on the domain on which 𝑓(𝑥)f(x) is shown.
(c) How many critical points does 𝐹(𝑥)F(x) have?
A. 0
B. 1
C. 2
D. 3
E. more than 3

(d) How many local maxima does 𝐹(𝑥)F(x) have?
A. 0
B. 1
C. 2
D. 3
E. more than 3

(e) How many local minima does 𝐹(𝑥)F(x) have?
A. 0
B. 1
C. 2
D. 3
E. more than 3

(f) How many inflection points does 𝐹(𝑥)F(x) have?
A. 0
B. 1
C. 2
D. 3
E. 4

 

Q4. The vertical velocity of a cork bobbing up and down on the waves in the sea is shown below. In this graph, upward is considered positive.

Sketch a graph of the height of the cork above the sea floor as a function of time, and use that to select the correct answers to complete the following statements.

 

A: At point A, the cork is at the midline of its motion, moving upwards.

At point B, the cork is at its highest point, moving neither upwards nor downwards.

At point C, the cork is at the midline of its motion, moving downwards.

At point D, the cork is at its lowest point, moving neither upwards nor downwards.

Q5. The graph of 𝑓″ is given in the figure below.

Draw graphs of 𝑓and 𝑓′, assuming both go through the origin, and use them to complete the following statements 

A. 𝑓(𝑥) is greatest at 𝑥= x3
B. 𝑓(𝑥) is least at 𝑥= x5
C. 𝑓′(𝑥) is greatest at 𝑥= x1
D. 𝑓′(𝑥) is least at 𝑥= x5
E. 𝑓″(𝑥) is greatest at 𝑥= x3
F. 𝑓″(𝑥) is least at 𝑥= x5