Mathematics

Problem

For differentiable functions defined parametrically by

x=f(t)

and

y=g(t)

, answer the following. a. (i) Starting from

x=f(t)

and

y=g(t)

, derive an expression for

\dfrac{dy}{dx}

in terms of

t

. State clearly any conditions required for the formula to hold. b. (ii) The curve

C

is given by

x=3t-2,\; y=t^{2}+1

. Find the equation of the tangent to

C

at the point corresponding to

t=2

. Write the answer in slope–intercept form. c. (iii) For the same curve

C

, determine all real values of

t

for which the tangent is parallel to the line

y=4x+7

. Hence obtain the equation of every such tangent.