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Math 1431

Section 12485 MWF 12‐1pm SR 117

Dr. Melahat Almus

almus@math.uh.edu

http://www.math.uh.edu/~almus

COURSE WEBSITE:

http://www.math.uh.edu/~almus/1431_sp16.html

Visit my website regularly for announcements and course material!

If you e-mail 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.

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.

DID YOU RESERVE A SEAT FOR TEST 2?

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Popper #

Question# If 54 3f x x , 1f ' ?

a) 1 b) 5 c) 10 d) 20 e) None

Question# If 2 4f x cos x , 0f ' ?

a) 0 b) 1 c) 4 d) 8 e) None

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Section 2.4 – Implicit Differentiation

Question: If 2 2 1x y , what is dy

dx ?

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How do you differentiate implicitly?

Take the derivatives of both sides with respect to x.

Collect all dy

dx (or y' ) on one side.

Solve for dy

dx (or y' ).

Example: Find dy

dx if 3 2 35 4 5x x y y .

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Example: Find dy

dx if 2 2 5 6x xy y .

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Example: Given 2 2 7x xy y ,

a) Find the equation of the tangent line to the curve at the point 2 1, .

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b) Find the point(s) where the curve has a horizontal tangent line.

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Example: Given 4y sin xy , find the slope of the tangent line to the curve at

(0,4).

Exercise: Given 2 1sin x cos y , find dy

dx.

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Example: Given 2 2 16x y , find 2

2d y

dx.

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Chapter 3

Section 3.1 – Related Rates

The Derivative as a Rate of Change

Given y f x , the slope of the secant line

f b f a

b a

gives the average rate of change in y with

respect to x over the interval a,b .

On the other hand, the slope of the tangent line, f ' a , gives the instantaneous

rate of change in y with respect to x at a . The instantaneous rate of change is the

limit of the average rates of change as the interval width approaches 0.

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Example: The side length of a square is changing.

The rate of change of the area with respect to side is:

How fast is the area changing when 5x ?

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Example: The radius of a right cylinder is decreasing while the height remains constant. What is the rate of change of volume with respect to radius?