1. Clicker Question 1 Solve for x : 5 + 3 (x+2) = 12 A. x = ln(12)/ln(8) – 2 B. x = ln(7/3) – 2 C. x = ln(7)/ln(3) – 2 D. x = ln(7) – ln(3) – 2 E. x = (ln(7) – 2)/ln(3)

2. Clicker Question 2 What is the value of sin(arctan(-1))? A. 0 B. -1 C. -  /4 D.  2 / 2 E. -  2 / 2

3. Clicker Question 3 Solve for t : cos(3t -2)=.4 A. t = (arccos(.4) + 2) / 3 B. t = (cos(.4) + 2) / 3 C. t = (arccos(.4) + 3) / 2 D. t = arccos(.8) E. t = (arccos(.8) + 2) / 3

4. Limits (2/4/09) Question: How can we compute the slope of the tangent line to the curve y = x 2 at the point (1, 1)? Possible approach: Compute the slope of the secant line which the connects the points (1, 1) and (1 + h, (1+ h ) 2 ) for small values of h. Now try to see the limit as h goes toward 0.

5. Instantaneous Velocity That example was the “Tangent Problem”. Now comes the “Velocity Problem”. Question: Given the position of a moving car as a function of time, how can we compute the “instantaneous velocity ” of the car at a specific moment? Possible approach: Compute the average velocity over a short period of time, and find the limit as that period approaches zero.

6. Limit of a Function at a Point In both problems above, we seek the limit of some function (often, but not always, the function is in the form of a ratio) as we approach some point. Definition: We say the limit as x approaches a of f (x) is a number L, writing lim x  a f (x) = L, if f ‘s values get closer and closer to L as x gets closer and closer to a.

7. Some Examples of Limits Some limits are obvious: lim x  3 x 2 = lim t   cos(t) = But some limits aren’t: lim t  0 sin(t) / t = lim h  0 ((3 + h) 2 - 9) / h = What was “problematic” about these two?

8. Clicker Question 4 What is lim x  8 (e x – 8 + log 2 (x ))? A. e + 2 B. 3 C. e + 3 D. 4 E. Does not exist

9. Clicker Question 5 What is lim x  3 (x 2 – 9) / (x – 3) ? A. 3 B. 6 C. 0 D. 1 E. Does not exist

10. Other Not Obvious Limits What is lim x  4 (x 2 – 3x - 4)/(x – 4) ? What is lim x  2 3/(x - 2) 2 ? What is lim x  1/x ? Note that in the last two examples, we are allowing the idea of infinity to be involved in limits, either as the answer (meaning the output keeps getting bigger and bigger) or as what x approaches (meaning x gets bigger and bigger).

11. Assignment for Friday Hand-in #1 is due next Tuesday. Here we go with calculus! Read Sections 2.1 and 2.2. In Section 2.1, do Exercises 3 and 5. In Section 2.2, do Exercises 1, 3, 5, 9, 15, 21, 25, and 27.