Page 20 - Calc for the Clueless

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If f(x) = 3x + 4x - 5 were a curve, the slope of the tangent line at any point on the curve would be found by
multiplying the x value by 6 and adding 4.

Example 3 Continued—

We don't need to use the 34.

Find the slope of the tangent line to the curve f(x) = 3x + 4x - 5 at the point (3,34).
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The slope m = 6x + 4 = 6(3) + 4 = 22.

Example 3 Continued, Continued—

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Find the equation of the line tangent to f(x) = 3x + 4x - 5 at the point (3,34).
From algebra, the equation of a line is given by

(point-slope). x 1 = 3, y 1 = 34, and the slope m is f'(3) = 22. The equation of the line is

which you can simplify if forced to.

Example 3 Last Continuation—

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Find the equation of the line normal to y = 3x + 4x - 5 at the point (3,34).
The word normal means draw the tangent line at point P and then draw the perpendicular to that tangent line at
point P. Perpendicular slope means negative reciprocal. The equation of the normal line is...

Example 4—

Find f'(x) using the definition of derivative.