Mathematics

Problem

Let

f(x)=x^{2}+kx

where

k

is a real parameter and let

a

be a fixed real number. a. (i) Obtain a simplified form of

f(a+h)-f(a)

. b. (ii) Evaluate

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

in terms of

a

and

k

. Keep your answer fully simplified. c. (iii) If the slope of the tangent to

y=f(x)

x=3

equals

14

, determine the value of

k