2.1: The Derivative and the Tangent Line Problem
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1 .1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position o te secant line as te second point approaces te irst. Deinition: Te tangent line to te curve y ( x ) at te point ( a, ( a )) is te line troug P wit slope m ( x) ( a) provided tis it exists. Equivalently, m ( ) ( ) 0 a a provided tis it exists. Note: I te tangent line is vertical, tis it does not exist. In te case o a vertical tangent, te equation o te tangent line is. Note: Te slope o te tangent line to te grap o at te point (, ( )) a a is also called te slope o te grap o at. How to get te second expression or slope: Instead o using te points ( a, ( a )) and ( x, ( x)) on te secant line and letting, we can use ( a, ( a )) and ( a, ( a )) and let 0.
2 .1. Example 1: Find te slope o te curve te tangent line at tis point. y 4x 1 at te point (3,37). Find te equation o Example : Find an equation o te tangent line to te curve 3 y t te point (1,1). Example 3: Determine te equation o te tangent line to ( x) t te point were x.
3 .1.3 Te derivative: Te derivative o a unction at x is te slope o te tangent line at te point ( x, ( x )). It is also te instantaneous rate o cange o te unction at x. Deinition: Te derivative o a unction at x is te unction ' wose value at x is given by '( x) ( ) ( ), provided tis it exists. 0 x x Te process o inding derivatives is called dierentiation. To dierentiate a unction means to ind its derivative. Equivalent ways o deining te derivative: '( x) ( ) ( ) x x x x x 0 (Our book uses tis one. It is identical to te deinition above, except uses x in place o.) '( x) ( w) ( x) w x w x '( a) ( x) ( a) (Gives te derivative at te speciic point were.) '( a) 0 ( a ) ( a) (Gives te derivative at te speciic point were.) Example 4: Suppose tat x 6x g( x ). Determine g '( x ) and 3 g '(3).
4 .1.4 Example 5: were x. Suppose tat ( x) x 1. Find te equation o te tangent line at te point x Example 6: Determine te equation o te tangent line to ( x) x 1 at te point, 4 5.
5 .1.5 Summary: Te slope o te secant line between two points is oten called a dierence quotient. Te dierence quotient o at a can be written in eiter o te orms below. ( x) ( a) ( a ) ( a). Bot o tese give te slope o te secant line between two points: ( x, ( x )) and ( a, ( a)) or, alternatively, ( a, ( a )) and ( a, ( a )). Te slope o te secant line is also te average rate o cange o between te two points. Te derivative o at a is: 1) te it o te slopes o te secant lines as te second point approaces te point ( a, ( a )). ) te slope o te tangent line to te curve y ( x ) at te point were. 3) te (instantaneous) rate o cange o wit respect to t a. 4) ( x) ( a) (it o te dierence quotient) 5) 0 ( a ) ( a) (it o te dierence quotient) Common notations or te derivative o y ( x ) : d '( x ) ( x ) dx y ' D ( x ) x dy dx D ( x) Te notation dy dx was created by Gottried Wilelm Leibniz and means dy x dx x 0 y. To evaluate te derivative at a particular number a, we write '( a ) or dy dx x a
6 .1.6 Dierentiability: Deinition: A unction is dierentiable at a i '( a ) exists. It is dierentiable on an open interval i it is dierentiable at every number in te interval. Teorem: I is dierentiable at a, ten is continuous at a. Note: Te converse is not true tere are unctions tat are continuous at a number but not dierentiable. Note: Open intervals: ( a, b ), (, a ), ( a, ), (, ). Closed intervals: [ a, b ], (, a ], [ a, ), (, ). To discuss dierentiability on a closed interval, we need te concept o a one-sided derivative. Derivative rom te let: ( x) ( a) Derivative rom te rigt: ( x) ( a) For a unction to be dierentiable on te closed interval [ a, b ], it must be dierentiable on te open interval ( a, b ). In addition, te derivative rom te rigt at a must exist, and te derivative rom te let at b must exist. Ways in wic a unction can ail to be dierentiable: 1. Sarp corner. Cusp 3. Vertical tangent 4. Discontinuity
![Note: Open intervals: ( a, b ), (, a ), ( a, ), (, ). Closed intervals: [ a, b ], (, a ], [ a, ), (, ). To discuss dierentiability on a closed interval, we need te concept o a one-sided derivative.](/docs-images/46/21159183/images/page_6.jpg "Derivative rom te let: ( x) ( a) Derivative rom te rigt: ( x) ( a) For a unction to be dierentiable on te closed interval [ a, b ], it must be dierentiable on te open interval ( a, b ).")
7 .1.7 Example 7: Example 8: Sketc te grap o a unction or wic (0), '(0) 1, () 1, 1 '(), '(3) '(), and '(5) 0. 3
8 .1.8 Example 9: Use te grap o te unction to draw te grap o te derivative. Example 10: Use te grap o te unction to draw te grap o te derivative.
9
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Chapter 2 The Derivative Business Calculus 74 Section 1: Instantaneous Rate of Change and Tangent Lines Instantaneous Velocity Suppose we drop a tomato from the top of a 100 foot building and time its