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Calculus Help sites:

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Calculus Help sites:

Websites of help:

Algebra type help:

www.purplemath.com

www.coolmath.com

Calculus help:

http://www.khanacademy.org/

Barron’s AP Calculus

5 Steps to a 5: AP Calculus AB/BC

Calculus Notes 3.1: Derivatives.

Definition: Slope of the tangent to a curve with equation y=f(x) at the point where x=a to be:

Start up:

1. What is the instantaneous rate of change of at

Definition: The derivative of a function f at a number a, denoted by f`(a), is

if the limit exists.

Definition: The tangent line to the curve at

is the line through with slope equal to , the derivative of f at a.

Instantaneous Rate of Change:

Definitions:

Also:

The derivative f`(a) is the instantaneous rate of change of y=f(x) with respect to x when x=a.

Example 2: Find f’(a)& f’(3)

Calculus Notes 3.1: Derivatives.

Example 1: For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning:

Negative slope

zero slope

positive slope

positive slope, but steeper than the one at g`(-1.2)

Calculus Notes 3.1: Derivatives.

Example 3: The fuel consumption (measured in gallons per hour) of a car traveling at a speed of v miles per hour is c=f(v).

(a) What is the meaning of the derivative f’(v)? What are its units?

f’(v) is the rate at which fuel consumption is changing with respect to the speed. Its units are (gal/hour)/(mi/hour)---gph/mph

(b) Write a sentence (in layman’s terms) that explains the meaning of the equation f’(20)=-0.05.

The fuel consumption is decreasing by 0.05 (gal/h)/(mi/h) as the car’s speed reaches 20 mi/h. So if you increase your speed to 21 mi/h, you could expect to decrease your fuel consumption by about 0.05 (gal/h)/(mi/h).

Calculus Notes 3.1: Derivatives.

Example 4: Life expectancy improved dramatically in the 20th century. The table gives values of E(t), the life expectancy at birth (in years) of a male born in the year t in the United States. Interpret and estimate the value of E’(1910) and E’(1950).

This means that life expectancy at birth was increasing at about 0.345 year/year in 1910.

This means that life expectancy at birth was increasing at about 0.205 year/year in 1950.

PS 3.1 pg.132 #3, 4, 6, 9, 13, 14, 18, 19, 27, 28, 31, 33, 35 (13)