Chapter 1. Calculus Tutorial 2.6.006

1.1 Problem Statement

{6,7,8,9}

Plot the graphs of

and

on the same set of axes. Compute if f(x) is squeezed by l(x) and u(x) at .

1.2 Step 1

Recall what it means for f(x) to be squeezed by l(x) and u(x) and how we can apply the Squeeze Theorem to find limits.

Question 1.1

The function f(x) is squeezed at x = c if there exist two functions l(x) and u(x) such that l(x) ≤ f(x) ≤ u(x) for all x ≠ c in an open interval containing c. Then if

we must have

M1jvgaUwpgE=.

This is called the Squeeze Theorem.

Incorrect.
Correct.

1.3 Step 2

Question Sequence

Question 1.2

Note that is always 05X2T0KwWYApuO7m9Uru25yxLm+uRMeNQ/dfd82IRE1vVhPwN49W7DiOtrFgmXF5aA4RziYnf99Vl9AIMSX3ntOe2o1yEogakCqTXg== zero.

Based on the graphing techniques of shifting, stretching and reflecting, it has the shape of an absolute value graph y = |x| shifted π/2 units to the y/8elwqLKvMQPfWtffwxDQ== and $a units Ls7sjBKmfE4GvFTi.

Incorrect.
Correct.

Question 1.3

The graph of has amplitude nc1ItEz0kR4=. Choosing the interval [0, π], reaches its maximum value at x = RyFOFx/iICSLzK5VUk1A4g== .

Incorrect.
Correct.

Question 1.4

/6tN7ULh5sQwakLZ7YYIgdLudFbQQZMc8yeIdX0QY1u2j1ULkjUcAqw/7ja85/hl+bBSHV2BIrexQTGR9r9pTG5jdGKAldGWIO6Yl5LInoqiXKvb
A.
B.
C.
D.
Incorrect.
Correct.

1.4 Step 3

Use the Squeeze Theorem to find if f(x) is squeezed by l(x) and u(x) at x = π/2.

In order to use the Squeeze Theorem, we would need to show that . Then by the Squeeze Theorem, would also be equal to L.

Question Sequence

Question 1.5

Either from the graph in Step 2, or the continuity of the functions, find and .

nc1ItEz0kR4=

nc1ItEz0kR4=

Incorrect.
Correct.

Question 1.6

Therefore, by the Squeeze Theorem, we can determine .

nc1ItEz0kR4=.

Incorrect.
Correct.