Tangent Line Recall from geometry Tangent is a

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Tangent Line • Recall from geometry • Tangent is a line that touches the

Tangent Line • Recall from geometry • Tangent is a line that touches the circle at only one point • Let us generalize the concept to functions • A tangent will just "touch" the line but not pass through it • Which of the above lines are tangent?

A Secant Line • Crosses the curve twice x=a h • The slope of

A Secant Line • Crosses the curve twice x=a h • The slope of the secant will be

Tangent Line • Now let h get smaller and smaller x=a h The slope

Tangent Line • Now let h get smaller and smaller x=a h The slope of the tangent line 3

The Derivative • We will define the derivative of f(x) as • Note •

The Derivative • We will define the derivative of f(x) as • Note • The derivative is the rate of change function for f(x) • The derivative is also a function of x • The limit must exist 4

Comparison Difference Quotient • Derivative • • Slope of secant • Slope of tangent

Comparison Difference Quotient • Derivative • • Slope of secant • Slope of tangent • Average rate of change • Average velocity • Instantaneous rate of change • Instantaneous velocity 5

(La. Grange’s Notation) “f prime of x” “the derivative of f with respect to

(La. Grange’s Notation) “f prime of x” “the derivative of f with respect to x” “y prime” or (Leibniz’s Notation) or “the derivative of y with respect to x” or “the derivative of f of x”

Tangent Line • Given • Determine the slope of the tangent line at x

Tangent Line • Given • Determine the slope of the tangent line at x = 0 • Evaluate • Once you have the slope and the point • You can determine the equation of the line 8

Examples: Find the derivative of the following functions 1) 2) 3)

Examples: Find the derivative of the following functions 1) 2) 3)

Try It Out • Use the strategy to find the derivatives of these functions.

Try It Out • Use the strategy to find the derivatives of these functions.

Equation of the Tangent Line • We stated previously that once we determine the

Equation of the Tangent Line • We stated previously that once we determine the slope of the tangent

Tangent Line Write equations of tangent lines thru x = 1 for each function:

Tangent Line Write equations of tangent lines thru x = 1 for each function:

Warning • Our definition of derivative included the phrase "if the limit exists" •

Warning • Our definition of derivative included the phrase "if the limit exists" • Derivatives do not exist at "corners" or "sharp points" on the graph • The slope is different on each side of the point • The limit does not exist f(x) = | x – 3 |