Section MWF 12 1pm SR 117

Transcription

1 Math 1431 Section 1485 MWF 1 1pm SR 117 Dr. Melahat Almus almus@math.uh.edu COURSE WEBSITE: Visit my website regularly for announcements and course material! Read the syllabus (posted on the course website!). If you me, please mention your course (1431) in the subject line. Be considerate of others in class. Respect your friends and do not distract anyone during the lecture. Check your CASA account for quiz due dates; don t miss any quizzes. Purchase popper scantrons from UH Bookstore. Bring one scantron to every class. BUBBLE IN PS ID VERY CAREFULLY! If you make a bubbling mistake, your scantron will not be saved in the system and you will not get credit for it even if you turned it in. Bubble in Popper Number. 1

2 Popper # 01 Question# Compute sin 8 0 a) 8 b) 4 c) d) 0 e) DNE Question# 0 sin 5 a) /5 b) 4/5 c) /5 d) 4/5 e) 0 Section 1.6 Pinching Theorem, Limits of Trig Functions In general, sin sinc and cosc cosc c c. Fact: sin 0 1 and 1 cos 0 0. Fact: sin a 0 b a b and 1 cos a 0 b 0. ALWAYS pay attention to what value is approaching.

3 3 Eamples: 0 5 h sin h h sin 0 4 cos Eamples: tan tan tan

4 tan 0 4 Eample: 1 cos 0 7 Eample: csc5 0 Eample: 0 1 cos 4 4

5 Eample: csc sin 0 Eample: sin 5t t0 tcos t POPPER# Question# tan 6 a) 0 b) 6 c) 3 d) 1 e) DNE NEXT: Chapter. 5

6 Chapter Section.1 The Definition of the Derivative What is a tangent line? What is a secant line? 6

7 Goal: Given a function, find the slope of the tangent line at the point a, f a. Let s try to approimate the slope of the tangent line at the point a, f a. 7

8 Pick another point Pa h,f a h on the curve and draw the secant line: The slope of this secant line is: f a h f a f a h f a aha h epression is called the difference quotient of f at a, f a.. This To make a better approimation, we need to move the two points closer together. In other words, we need to make h smaller. 8

9 Now, the slope of this new secant line is a better approimation for the slope of the tangent line. We want to let the distance between two points get smaller and smaller in order to minimize the error. That is, we need to let h approach 0 by taking its. This gives us a formula for finding the slope of the tangent line. Check this link: The slope of a tangent line to a function f at the point f a h f a h0 h if this it eists., a, f a is 9

10 Definition: A function f is said to be differentiable at a, if the it f a h f a h0 h eists. The value of this it is called the derivative of f at a and is denoted by f ' a. In other words, we define f' a provided that the it eists. f a h f a, h0 h f ' a is a real number and gives the slope of the tangent line at a, f a. This number is also said to be the slope of the graph of f at Note that a, f a. 10

11 f ' a is the slope of the tangent line at Remarks: 1. Since a, f a, using the point-slope equation of a line, we can write the equation of the tangent line: y f a f' a a.. f ' a is often referred to as the rate of change in f at a. Eample: Find the slope of the line tangent to the function f at the point 4,. Write the equation of the tangent line. 11