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Question

Find 𝑓(3) and 𝑓′(3), assuming that the tangent line to 𝑦 = 𝑓(𝑥)at 𝑎 = 3 has equation 𝑦= 9 𝑥+ 3. 

𝑓(3)=  

𝑓′(3)=

Answer 1

The slope of the tangent line at x=3 is 9, this is the same as f'(3)

Equation of tangent line at x=a is y = f(a) + f'(a) (x-a)

a = 3

f'(a) = f'(3) = 9 = slope of tangent line

9x + 3 = f(3) + f'(3) (x-3) = f(3) + 9 (x-3) = f(3) + 9x - 27

So f(3) - 27 = 3, giving f(3) = 30

f(3) = 30

f'(3) = 9