Question

Use the equation m Subscript PQ Baseline equals StartFraction f left parenthesis x 1 plus h right parenthesis minus f left parenthesis x 1 right parenthesis Over h EndFraction mPQ= fx1+hâfx1 h to calculate the slope of a line tangent to the curve of the function y equals f left parenthesis x right parenthesis equals 2 x squared y=f(x)=2x2 at the point Upper P left parenthesis x 1 comma y 1 right parenthesis equals Upper P left parenthesis 3 comma 18 right parenthesis Px1,y1=P(3,18).

Answer 1

m = 12

Explanation:

For f(x) = 2x² you want to find the slope m using ...

m = (f(x+h) -f(x))/h

at x=3.

Filling in the function definition and its arguments, we have ...

m = (2(x+h)² -2x²)/h = (2x² +4hx +2h² -2x²)/h = 4x +2h

The value of this as h goes to zero is ...

m = 4x

At x=3, the slope is ...

m = 4·3 = 12

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The attached graph shows a numerical estimate of the slope (f'(3)=12) and the line with that slope through the point P(3, 18). The line certainly appears to be tangent to the curve at that point.