Calculus problem

Question

Use the limit definition of the derivative to find the standard form equation of the tangent line to f(x) = 2x^2-5x-9 at x=3

Answer 1

The limit definition most convenient for this problem is probably

f'(x) = lim_{h → 0} (f(x + h) - f(x))/h

Thus when f(x) = 2x² - 5x - 9 and x = 3 we have

f'(3) = lim_{h → 0} (f(3 + h) - f(3))/h

= lim_{h → 0} ((2(3 + h)² - 5(3 + h) - 9) - (2(3)² - 5(3) - 9))/h

= lim_{h → 0} (12h + 2h² - 5h)/h

= lim_{h → 0} 7 + h²

= 7 + 0

= 7

Note a few skips were stepped in the algebra there -- it's recommended to work them out yourself to make sure everything checks out!