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2/3/2011

Math 103 – Rimmer 3.1/3.2 The

The of the of the secant li nes is the of the . secant line slope fx( ) − fa( ) m = lim x→ a x− a  f′( a ) the slope of the tangent line to fx() at x= a .

Math 103 – Rimmer 3.1/3.2 The Derivative Another expression for the slope of the tangent line. secant line slope fa( + h) − fa( ) m = lim  h→ 0 h f′( a ) the slope of the tangent line to fx() at x= a .

1 2/3/2011

Math 103 – Rimmer 3.1/3.2 The Derivative If you zoom in on the of tangency, the function is "locally linear" there.

Module 3.1 http://www.stewartcalculus.com/tec/

Math 103 – Rimmer List the following numbers from smallest to largest. 3.1/3.2 The Derivative

least+ steep

most steep + − +

g′(0) < 0 < g′(4) < g′(2)

2 2/3/2011

Math 103 – Rimmer Find the equation of the tangent line to a = − 1 3.1/3.2 The Derivative the graph of the function y= 2 x3 − 5 x at () − 1,3 . f (−1) fx( ) − fa( ) f( x) − f ( − 1) 2x3 − 5 x − 3 m = lim = lim = lim x→ a x− a x→ − 1 x −() − 1 x→ − 1 x +1 ( x+12)( x2 − 2 x − 3 ) −120 − 5 − 3 = lim x→ − 1 ()x +1 −2 2 3 =lim( 2x2 − 2 x − 3 ) 2 −2 −3 0 x→ − 1 x2 x const. =21()() −2 − 21 −− 3 = 1 Equation of the tangent line: m =1 y= mx + b ⇒ 3= 1( − 1 ) + b (−1,3 ) ⇒ b = 4 x y y= x + 4

Math 103 – Rimmer 3.1/3.2 The Derivative We could have used the other formula to get the slope of the tangent line. y=2 x3 − 5 x at ( − 1,3 )

fa( + h) − fa( ) f(−+1 h) − f ( − 1 ) m = lim = lim h→0 h h→0 h ()()()()−+1h3 =−+ 1 h −+ 1 h − 1 + h 3 3 fh()()()−+1 = 21 −+ h − 51 −+ h ()−+1h =−() 12 hh +2 () −+ 1 h × −1 h 3 2 f(−+=1 hhhh) 2( − 3 +−+− 3155) h 1 −1 h − − 2 f(−1+ h) = 2 hh3 − 6 2 + 6h−2 + 5 − 5 h 2h 2h 2 h h2 −h2 h 3 3 2 f(−+1 h) = 26 h − hh ++ 3 ()−1 +h3 = h3 − 3 h 2 + 3 h − 1

2 2h3− 6 h 2 ++− h 3 (3) 2h3− 6 h 2 + h h(2 h− 6 h + 1 ) m = li m = lim = lim h→0 h h→0 h h→0 h

=lim( 2h2 − 6 h + 1 ) = 1 h→0

3 2/3/2011

Math 103 – Rimmer 3.1/3.2 The Derivative

∆y fx( ) − fx( ) This is called a = 2 1 ∆x x2 − x 1 This is the average rate of change of y= f( x )

with respect to x over the [] x1 , x 2 .

∆y fx( ) − fx( ) lim= lim 2 1 ∆x →0 x → x ∆x2 1 x2 − x 1 The derivative f′( a ) is the instantaneous rate of change of yfx=() with respect to x when xa = .

Math 103 – Rimmer Interpreting the derivative as a rate of change. 3.1/3.2 The Derivative

The cost of producing x ounces of gold fr om a new gold mine is C( x ) dollars.

What is the meaning of C′( x ) ? What are its u nits? change in C∆ C C′() x measures the ratio: = change in x∆ x C′( x ) is the rate of change of production cos t with respect to the number of ounces produced, this is c alled marginal cost .

The units for C′( x ) are dollars per ounce.

What does C′( 800) = 17 mean ?

C′(800) is a ratio so let's turn 17 into a f raction.

17 C′()800 = 1 When you are producing 800 ounces of gold and you increase production by 1 to 801 ounces, cost will increase by $17.

4 2/3/2011

Math 103 – Rimmer 3.1/3.2 The Derivative

Let the number a vary.

fxh( +) − fx( ) f′() x = lim h→0 h

f′( x) can be thought of as a new function, it is called the derivative of f .

If fa′( ) exists, then f is called differentiable at a .

f is called differentiable on ( a , b ) if it is differentiable for all numbers in ()a , b .

Math 103 – Rimmer 3.1/3.2 The Derivative Find the derivative of the function using the definition of the derivative . fxh( +) − fx( ) fx( ) =4 x − 7 x 2 f′() x = lim h→0 h

fxh()()()+=4 xh +− 7 xh + 2 =+−447x h( x2 + 2 xh + h 2 ) fxh( +=+−) 4 x 4 h 7 x2 − 14 xh − 7 h 2 −fx( ) =−4 x + 7 x 2 fxh( +) − fx( ) = 4h− 14 xh − 7 h 2 =h(4 − 14 x − 7 h )

fxh( +) − fx( ) h(4− 14 x − 7 h ) f′() x = lim = lim =lim( 4 − 14x − 7 h ) h→0 h h→0 h h→0 f′() x=4 − 14 x

5 2/3/2011

Math 103 – Rimmer 3.1/3.2 The Derivative Find the derivative of the function using the definition of the derivative . 1 fxh( +) − fx( ) f() x = f′() x = lim x h→0 h 1 1 1 f() x+ h = fxh()()+− fx = − x+ h x+ h x 1 1 − x+ h x x x+ h x− x + h ( x+ x + h ) f′() x = lim ⋅ = lim ⋅ h→0 h x x+ h h→0 h xx()+ h ()x+ x + h x−( x + h ) −h = lim = lim h→0 hxxh()+() x + xh + h→0 hxxh()+() x + xh + −1 −1 −1 −1 = lim = = f′() x = h→0 xxh()+() x + xh + x2 () x+ x 2x x 2x3/2

Math 103 – Rimmer 3.1/3.2 The Derivative

6 2/3/2011

Math 103 – Rimmer 3.1/3.2 The Derivative Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV.

The main connection: function: sign of the slope of the tangent line derivative: + ⇒ above x− axis, − ⇒ below x − axis 0⇒ "touches" x − axis (a): sign of the slope of the tangent li ne −→0 →+→ 0 →− deriv.: below,then 0, then above, then 0 , then below

(a)⇔ II

(b): sign of the slope of the tangent li ne +→dne →−→ dne →+ deriv.: above, then jump to below, then jump to above (b)⇔ IV

(c): sign of the slope of the tangent li ne −→0 →+ deriv.: below,then 0, then above (c)⇔ I

(d): sign of the slope of the tangent li ne +→0 →−→ 0 →+→ 0 →− deriv.: above, then 0, then below, then 0, then above, then 0, then below

(d)⇔ III

Math 103 – Rimmer 3.1/3.2 The Derivative Animation of the graph of the derivative function http://www.stewartcalculus.com/tec/ Module 3.2

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