Calc2.43

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Find the slope of the tangent line to the graph f(x)=x^{2} when x=0,1,2,3,4,5,6.

This problem is just like past problems except that we are finding the slope of the tangent line at multiple points. First, find the derivative of f(x).

f'(x)=2x\,

Plugging in our values, we get

f'(0)=0\,

f'(1)=2\,

f'(2)=4\,

f'(3)=6\,

f'(4)=8\,

f'(5)=10\,

f'(6)=12\,

Our original graph is that of a parabola. One half of a parabola is increasing. Notice that it not only increases but increases at an increasing rate. You can see that by looking at the graph and now we can see that by looking at the derivative because the derivative gives us the slope of the tangent line or rate of change. Since the derivative is increasing, this indicates that the rate of change is increasing.


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