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Question

Let f(x)=1/2x2-1 and mPQ be the slope of the secant line that contains points P=(2,f(2)) and Q=(2+h,f(2+h)). Use question 3, and 4 and 5 determine a number a such that mPQ arrow a as h arrow 0. As your answer input the value of a.

Answer 1

Since the point P is (2, f(2)), first find the value of f(2).

f(x) = 1/2x² - 1

f(2) = 1/2(2)² - 1 = 1/2(4) - 1 = 2 - 1 = 1

So P = (2, 1)

We can arbitrarily select h = 1 so Q = (3, f(3)) = (3, 3.5) which means mPQ = (3.5 - 1)/(3 - 2) = 2.5

Then start to make h smaller. Now let h = 0.5 meaning Q = (2.5, f(2.5)) = (2,5, 2.125) which means mPQ = (2.125 - 1)/(2.5 - 2) = 2.25

Now let h = 0.1 meaning Q = (2.1, f(2.1)) = (2.1, 1.205) which means mPQ = (1.205 - 1)/(2.1 - 2) = 2.05

Now let h = 0.01 meaning Q = (2.01, f(2.01)) = (2.01, 1.02005) which means mPQ = (1.02005 - 1)/(2.01 - 2) = 2.005

Now let h = 0.001 meaning Q = (2.001, f(2.001)) = (2.001, 1.0020005) which means mPQ = (1.0020005 - 1)/(2.001 - 2) = 2.0005

Can you see that mPQ is approaching 2 as h approaches zero?

Answer 2

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