Chap3_Sec1

Chap3_Sec1 - of f at 0, that is, So f ( x ) = f (0) a x 1...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Compute the derivative of f(x) = x 8 + 12 x 5 – 4 x 4 + 10 x 3 – 6 x + 5 NEW DERIVATIVES FROM OLD Example 5 (3.1)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Find the points on the curve y = x 4 - 6 x 2 + 4 where the tangent line is horizontal. NEW DERIVATIVES FROM OLD Example 6 (3.1)
Background image of page 2
So, the given curve has horizontal tangents when x = 0, , and - . s The corresponding points are (0, 4), ( , -5), and (- , -5). NEW DERIVATIVES FROM OLD Example 6 3 3 3 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The equation of motion of a particle is s = 2 t 3 - 5 t 2 + 3 t + 4, where s is measured in centimeters and t in seconds. s Find the acceleration as a function of time. s What is the acceleration after 2 seconds? NEW DERIVATIVES FROM OLD Example 7 (3.1)
Background image of page 4
Let’s try to compute the derivative of the exponential function f ( x ) = a x using the definition of a derivative: 0 0 0 0 ( ) ( ) '( ) lim lim ( 1) lim lim x h x h h x h x x h h h f x h f x a a f x h h a a a a a h h + + - - = = - - = = EXPONENTIAL FUNCTIONS
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The factor a x doesn’t depend on h . So, we can take it in front of the limit: s Notice that the limit is the value of the derivative
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: of f at 0, that is, So f ( x ) = f (0) a x 1 '( ) lim h x h a f x a h -= 1 lim '(0) h h a f h -= EXPONENTIAL FUNCTIONS If f ( x ) = e x -x, find f and f . Compare the graphs of f and f . EXPONENTIAL FUNCTIONS Example 8 (3.1) The function f and its derivative f are graphed here. s Notice that f has a horizontal tangent when x = 0. s This corresponds to the fact that f (0) = 0 . EXPONENTIAL FUNCTIONS Example 8 Notice also that, for x > , f ( x ) is positive and f is increasing. When x < , f ( x ) is negative and f is decreasing. At what point on the curve y = e x is the tangent line parallel to the line y = 2 x ? EXPONENTIAL FUNCTIONS Example 9 (3.1) Thus, the required point is: ( a , e a ) = ( l n 2, 2) EXPONENTIAL FUNCTIONS Example 9...
View Full Document

Page1 / 10

Chap3_Sec1 - of f at 0, that is, So f ( x ) = f (0) a x 1...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online