Use the limit definition to compute f′(a) and find an equation of the tangent line. f(t)=t-10t^2, a=3

Answer 1

F(t+h) = (t+h) - 10(t+h)^2

= (t+h) [ 1 - 10(t+h)]

= (t+h) [ 1 - 10t - 10h]

F(t+h) - F(t) = (t+h) [ 1 - 10t - 10h] - t + 10t^2

= t - 10t^2 - 10th + h - 10th - 10h^2 - t + 10t^2

= -20th + h - 10h^2

dividing by h: -20t + 1 - 10h

as h tends to zero, the derivative is -20t + 1

which agrees with the power rule...

at x=3 , the slope is -59

f(3) = 3 - 90 = -87

so the tangent line has slope -59 and eats (3,-87)

B = y - mx = -87 - 3(-59)

= -87 + 177

= 90

y = -59x + 90