lecture17

# lecture17 - Math 006(Lecture 17 The Derivative dy...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 006 (Lecture 17) The Derivative dy Deﬁnition 1. For y = f (x), we deﬁne the derivative of f at x, denoted by f (x), or dx df , to be dx f (x + h) − f (x) f (x) = lim h→0 h if the limit exist. If f (x) exists for each x in the interval a < x < b, then f is said to be diﬀerentiable over a < x < b. Summary 1. Slope of the tangent line: f (x) is the slope of the line tangent to the graph of f at the point (x, f (x)). 2. Instantaneous rate of change: f (x) is the instantaneous rate of change of y = f (x) with respect to x. 3. Velocity: If f (x) is the position of a moving object at time x, then v = f (x) is the velocity of the object at that time. Procedure for ﬁnding the derivative Example 1. Find the derivative of f at x, for f (x) = 2x − x2 . Step 1: Find f (x + h) Step 2: Find f (x + h) − f (x) Step 3: Find f (x + h) − f (x) h Step 4: Find lim f (x + h) − f (x) h→0 h 1 Example 2. Find f (x) for each of the following functions: (a) f (x) = x3 (b) f (x) = (c) f (x) = 1 x √ x Basic Diﬀerentiation Properties Theorem 1. (Power Rule) If y = f (x) = xn where n is a real number, then f (x) = nxn−1 . Example 3. Find f (x) for each of the following functions: (a) f (x) = 3 (b) f (x) = x5 (c) f (x) = x3/2 (d) f (x) = x−3 1 (e) f (x) = √ 3 x (f) f (x) = x 2 . √ 2 Theorem 2. If y = f (x) = ku(x), where k is a constant, then f (x) = ku (x). If y = f (x) = u(x) ± v (x), then f (x) = u (x) ± v (x). Example 4. Find the derivative of (a) f (x) = 3x4 − 2x3 + x2 − 5x + 7 (b) f (x) = 3 − 5 x2 (c) y = 6v 4 − √ 5 v (d) d dx 1 3 x2 +√ − 5x4 2 x (e) f (x) = x2 + 25 x2 3 Example 5. An object moves along the y axis so that its position at time x is f (x) = x3 − 6x2 + 9x (a) Find the instantaneous velocity function v . (b) Find the velocity at x = 2 and x = 5. (c) Find the time(s) when the velocity is 0. Example 6. Let f (x) = 2x3 − 9x2 + 12x − 54. (a) Find f (x). (b) Find the equation of tangent to the graph of y = f (x) at x = 0. (c) Find the value(s) of x where the tangent to the graph of y = f (x) at x is horizontal. 4 ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

lecture17 - Math 006(Lecture 17 The Derivative dy...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online