I\'m confused as to how I justify this. Please explain your answer in depth. Tha

Question

I'm confused as to how I justify this. Please explain your answer in depth.

Thanks.

Let f(x) = 2013x. Justify why for any real number a the following identity hold: f(x) = 2013af(x - a). Knowing that the slope of the tangent line to the curve y = 2013x at (0,1) is (approximately) 7.6, use the geometric information suggested by the previous part to compute the equation of the tangent line to the curve y = 2013x at the point (a,2013a). Use the geometric information suggested by the previous part to compute the equation of the tangent line to the curve y = log2013 x at the Point (a, log2013 ). (Careful, a can be negative).

Explanation / Answer

f(x) = 2013^x

f(x-a) = 2013 ^(x-a) =- 2013^x / 2013^a = f(x) / 2013^a

Thus, f(x-a) = f(x) / 2013^a which implies : f(x) = 2013^a f(x-a)

Now, the equation of tangent at point (a, 2013^a) shall be given as:

(y - 2013^a) = m (x - a)

Now, m = dy/dx = 2013^a (log 2013) = 7.6 ( 2013^a)

So, equation of tangent at (a,2013^a) is

y = 2013^a + 7.6(2013^a)(x-a)

C) The tangent slope to curve: Y = log |x| / log (2013) by changing the base to 'e'

slope = dy/dx = 1 /7.6x

So, equation of the tangent =

(y - ln|a|/7.6 ) = [ x - a ] / 7.6a

Hope this helps. Please rate it the best. Ask doubts if you have any. thanks.

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