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 to be squeezed by l and u and how we can apply the Squeeze Theorem to find limits.

Question 1.1

The function f is squeezed at x = c if there exist two functions l and u 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.