Question

If f(2)=1 and f(2+h)=(h+1)3, compute f′(2).

If f(−1)=5 and f(−0.9)=5.2, estimate f′(−1).

If the line y=−3x+2 is tangent to f(x) at x=−4, find f(−4). Your answer should be expressed as an integer.

If the line y=x−1 is tangent to the graph of f(x) at x=−3, find f′(−3).

Answer 1

Use the definition of a derivative:
f'(x) = lim_{∆x→0} (f(x+∆x)-f(x))/∆x

Like so:
f(2) = 1
f(2+h) = (h+1)^3

f'(2) = lim_{h→0} (f(2+h)-f(2))/h
f'(2) = lim_{h→0} ((h+1)^3 -1)/h
f'(2) = lim_{h→0} (h^3 + 3h^2 + 3h)/h
f'(2) = lim_{h→0} (h^2 + 3h + 3)
f'(2) = 3

Answer 2

Thanks, can you tell me the rest?