91Ó°ÊÓ

The graph of \(f(x)=\frac{a x+b}{(x-1)(x-4)}\) has a horizontal tangent line at (2,-1) Find \(a\) and \(b\). Check using a graphing calculator.

Find \(h^{\prime}(2),\) given \(h(x)=f(g(x)), f(u)=u^{2}-1, g(2)=3,\) and \(g^{\prime}(2)=-1\)

Show that the line \(4 x-y+11=0\) is tangent to the curve \(y=\frac{16}{x^{2}}-1\).

Determine the point on the parabola \(y=-x^{2}+3 x+4\) where the slope of the tangent is \(5 .\) Illustrate your answer with a sketch.

A football is kicked up into the air. Its height, \(h,\) above the ground, in metres, at \(t\) seconds can be modelled by \(h(t)=18 t-4.9 t^{2}\) a. Determine \(h^{\prime}(2)\) b. What does \(h^{\prime}(2)\) represent?

The concentration, \(c,\) of a drug in the blood \(t\) hours after the drug is taken orally is given by \(c(t)=\frac{5 t}{2 t^{2}+7} .\) When does the concentration reach its maximum value?

Determine the coordinates of the points on the graph of \(y=x^{3}+2\) at which the slope of the tangent is 12.

A 50000 L tank can be drained in 30 min. The volume of water remaining in the tank after \(t\) minutes is \(V(t)=50000\left(1-\frac{t}{30}\right)^{2}, 0 \leq t \leq 30 .\) At what rate, to the nearest whole number, is the water flowing out of the tank when \(t=10 ?\)

For the function \(f(x)=x|x|,\) show that \(f^{\prime}(0)\) exists. What is the value?

The position from its starting point, \(s\), of an object that moves in a straight line at time \(t\) seconds is given by \(s(t)=\frac{t}{t^{2}+8} .\) Determine when the object changes direction.