Here, we are required to find the slope of the tangent line to the graph of the given function at any point.
The correct answer is 16x + 3.
The four step process to find the slope of the tangent line to the graph at any point is as follows;
First step (1):
- First, there's a need to evaluate the value of F(x+h), the result of which is;
f(x +h) = 8(x + h)² + 3( x + h) and yields;
f(x + h) = 8( x² + 2xh + h²) + 3( x + h)
and, f (x + h) = 8x² + 16xh + 8h² + 3x + 3h.
Second step(2):
- Second step involves subtraction of F(x) from F(x+h), i.e
F(x+h) - f(x) = 8x² + 16xh + 8h² + 3x + 3h - ( 8x² + 3x)
and, F(x+h) - f(x) = 16xh + 8h² + 3 h
Third step (3):
- The third step involves dividing both sides of the equation by h, i.e
{F(x+h) - f(x)} / h = (16xh)/h + (8h²)/h + (3 h)/h
This in turn yields;
{F(x+h) - f(x)} / h = 16x + 8h + 3.
Fourth step(4):
- This step involves taking limits on both sides as h => 0., i.e
lim f(x + h) - f(x)/ h =
h=>0
lim 16x + 8h + 3 = lim 16x + 8(0) + 3 = 16x + 3
h=>0. h=>0
Therefore, the slope of the tangent line to the graph of the given function at any point is;
16x + 3.
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