Let f(x) = 4x^2 - 3x + 2. Recall the derivative of f at x = a, which is denoted
ID: 2863929 • Letter: L
Question
Let f(x) = 4x^2 - 3x + 2. Recall the derivative of f at x = a, which is denoted by f'(x). is the slope of the tangent line to the graph of y = f(x) and. in terms of the limits of the slopes of the secant lines. Is f'(a) = lim_x rightarrow a f(x) - f(a)/x - a = lim_h rightarrow 0 f(a + h) - f(a)/h. Find a formula for f'(a) by computing the limit both ways. i.e., as a limit as x rightarrow a and as a limit as h rightarrow 0. Write out your solutions nicely on a clean sheet of paper, and take these solutions to your next recitation, where they may be contacted and graced. f'(a) =Explanation / Answer
f(x) = 4x^2 - 3x+2
f(a) = 4a^2 - 3a + 2
f'(a) = f(x) - f(a)/(x-a)
So, f'(a) = (4x^2 - 3x+2 - 4a^2 + 3a -2)/(x-a)
f'(a) = 4(x^2 - a^2) -3(x-a))/(x-a)
f'(a) = 4(x+a)(x-a) - 3(x-a))/(x-a)
So, f'(a) = 4x + 4a - 3
Using part 2,
f(a+h) = 4a^2+4h^2+8ah - 3a - 3h + 2
f(a) = 4a^2 - 3a + 2
As h tends to 0,
f(a+h) - f(a)/h
On solving,
we get
8a - 3
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