Let (f(x)=x^ 3 -4x )

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Let

f(x)=x^{3}-4x

. The point

(2, f(2))

lies on the graph of

f

. (a) Write an explicit algebraic expression for

\displaystyle \frac{f(2+h)-f(2)}{h}

and simplify the numerator completely. (b) Evaluate

\displaystyle \lim_{h\to 0}\frac{f(2+h)-f(2)}{h}.

Show each factorization and cancellation step. (c) Give the numerical value of the instantaneous rate of change of

f

x=2

. Interpret this value as the slope of the tangent line at the point

(2,f(2))

. Then determine the equation of that tangent line in point–slope form. (d) Suppose

\Delta x

is a small positive number. Use your result from part (c) to produce a linear approximation for

f(2+\Delta x)

, and briefly comment on when this approximation will be most accurate.