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Calculus Help sites: Websites of help: Algebra type help: Calculus help: Barron’s AP Calculus 5 Steps to a 5: AP Calculus AB/BC

Calculus Notes 3.1: Derivatives. Start up: 1. What is the instantaneous rate of change of at Definitions: Definition: Slope of the tangent to a curve with equation y=f(x) at the point where x=a to be: Definition: The derivative of a function f at a number a, denoted by f`(a), is if the limit exists. Also: Definition: The tangent line to the curve at is the line through with slope equal to, the derivative of f at a. The derivative f`(a) is the instantaneous rate of change of y=f(x) with respect to x when x=a. Instantaneous Rate of Change:

Example 1: For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning: Calculus Notes 3.1: Derivatives. Negative slope zero slope positive slope positive slope, but steeper than the one at g`(-1.2) Example 2: Find f’(a)& f’(3)

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). Calculus Notes 3.1: Derivatives. (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)= 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).

PS 3.1 pg.132 #3, 4, 6, 9, 13, 14, 18, 19, 27, 28, 31, 33, 35 (13) Calculus Notes 3.1: Derivatives. Example 4: Life expectancy improved dramatically in the 20 th 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). tE(t) This means that life expectancy at birth was increasing at about year/year in This means that life expectancy at birth was increasing at about year/year in 1950.