Calculus I, Pre-Lab 8: Related Rates, Differentials and ...



Calculus I, Pre-Lab 10: Inverse Trig Functions, Hyperbolic Functions and Extreme Values

Name:

The Pre-lab is intended to encourage you to prepare for the Lab, so answer these questions in your own words, and hand this sheet in at the beginning of lab.

1. Read activity #1 in Lab 10 and section 3.6 in the text. In the activity, you are referred to problem 25 page 234. The instructions for this problem ask you to use the method leading to Formula 3. What page is Formula 3 on?

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Let's take the derivation for Formula 3 as a template: (fill in the ten question marks)

[pic]

Just like in problem 25 where you needed advance knowledge of the derivative of the cos function, in problem #27, you will need to have advance knowledge of the derivative of the cotangent function.

On what page of the textbook is this formula enunciated?

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In the right triangle below, notice that the cotangent of the angle y is equal to [pic]. Use the Pythagorean Theorem to find the length of the hypotenuse in terms of x. (Put your answer by the hypotenuse in the figure.) Use your completed triangle to figure out [pic] in terms of [pic].

[pic]

2. Read Lab 10#2. As mentioned at the bottom of page 236, the formulas in the box at the top of page 237 are analogous to differentiation rules for trigonometric functions.

(a) On what page of the textbook is the analogous box showing the derivatives of the trig functions?

(b) Have you memorized these analogous differentiation rules for the trig functions?

Look at the table of differentiation rules at the bottom of page 131.

(c) Which one will you need to use when figuring out the derivative of the hyperbolic tangent function?

(d) On what page of the textbook does the author figure out the derivative of the tangent function?

3. Read section 4.1. Look at FIGURE 4 on page 255.

(a) Is the slope of the tangent line to the graph of [pic] equal to zero at the absolute maximum?

(b) Is the slope of the tangent line to the graph of [pic] equal to zero at the absolute minimum?

Look at FIGURE 10 page 258.

(c) What is the slope of the graph at the place where the absolute minimum occurs?

Look at FIGURE 11page 258.

(d) At what points in the domain of f is slope equal to zero?

(e) What is the absolute maximum value of f on its domain?

(f) What is the absolute minimum value of f on its domain?

4. In section 4.1, you will find three boxed in definitions-see boxes (1), (2) and (6). Imagine trying to write the section without making any definitions at all, and then think about why the author put these definitions in. Write down two good reasons for putting these definitions in.

REASON 1:

REASON 2:

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