Mathematics

Problem

Consider the polynomial function

f(x)=x^{3}+2x^{2}-5

. a. (i) Simplify the expression

f(1+h)-f(1)

. b. (ii) Hence evaluate the limit

\displaystyle\lim_{h\to 0}\frac{f(1+h)-f(1)}{h}

. Write the final value as a single integer. c. (iii) What is the slope of the tangent to the curve

y=f(x)

at the point

(1,\,f(1))

? State your answer clearly.