# In Example 6 in this section, we were given f ( x ) = 3 x 2 + 2 x + 11 and found f ' ( x ) = 6 x + 2 . Find (a) the instantaneous rate of change of f(x) at x = 6. (b) the slope of the tangent to the graph of y = f(x) at x = 6. (c) the point on the graph of y = f(x) at x = 6. EXAMPLE 6 Tangent Line Given y = f ( x ) = 3 x 2 + 2 x + 11 , find (a) the derivative of f(x) at any point ( x, f ( x )). (b) the slope of the tangent to the curve at (1, 16). (c) the equation of the line tangent to y = 3 x 2 + 2 x + 11 at (1.16). Solution (a) The derivative of f ( x ) at any value x is denoted by f'(x) and is y ' = f ' ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 [ 3 ( x + h ) 2 + 2 ( x + h ) + 11 ] − ( 3 x 2 + 2 x + 11 ) h = lim h → 0 3 ( x 2 + 2 x h + h 2 ) 2 x + 2 h + 11 − 3 x 2 − 2 x − 11 h = lim h → 0 6 x h + 3 h 2 + 2 h h = lim h → 0 ( 6 x + 3 h + 2 ) = 6 x + 2 (b) The derivative is f ' ( x ) = 6 x + 2 , so the slope of the tangent to the curve at (1,16) is f ' ( 1 ) = 6 ( 1 ) + 2 = 8 . (c) The equation of the tangent line uses the given point (1, 16) and the slope m = 8. Using y − y 1 = m ( x − x 1 ) gives y − 16 = 8 ( x − 1 ) , or y = 8 x + 8 . The discussion in this section indicates that the derivative of a function has several interpretations.

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781305108042

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781305108042

#### Solutions

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Chapter 9.3, Problem 8E
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