Differentiation: exam review Exam review: second paper

The graph of \(y=f(x)\), for \(0\le x\le 6.5\), has a local maximum at A\((2,52)\) and a local minimum at B\((5,25)\). [6]

1. (a) Write down the values of \(x\) for which \(f'(x)=0\). [1]
2. (b) Write down the interval on which \(f\) is decreasing. [2]
3. (c) Sketch the graph of \(f'(x)\) for \(0\le x\le 6.5\), showing where it crosses the \(x\)-axis. [3]

The diagram shows the graph of \(y=f'(x)\), the gradient function of a function \(f\). The graph crosses the \(x\)-axis at \(x=1\) and \(x=5\). [6]

1. (a) Write down the values of \(x\) at which \(f\) has a stationary point. [1]
2. (b) Write down the interval on which \(f\) is increasing. [2]
3. (c) For each stationary point, state whether it is a local maximum or a local minimum. Give a reason for each answer. [3]

A function is defined by \(f(x)=x^{4} - 9 x^{2}\) for \(-4\le x\le 4\). [6]

1. (a) Use your calculator to sketch the graph of \(y=f(x)\), showing the \(x\)-intercepts. [2]
2. (b) Write down the values of \(x\) for which \(f(x)>0\). [1]
3. (c) Find the values of \(x\) for which \(f'(x)<0\), that is, where \(f\) is decreasing. [3]

Differentiate each of the following with respect to \(x\). [6]

1. (a) \(f(x)=7x^4+\frac{8}{x^3}-3x+2\) [2]
2. (b) \(g(x)=(x+4)(2x-3)\) [2]
3. (c) \(h(x)=\dfrac{9 x^{4} - 6 x + 10}{x^{3}}\) [2]

The volume \(V\) litres of water in a tank \(t\) hours after a valve is opened is modelled by \(V=1.5t^2-18t+120\), for \(0\le t\le 12\). [6]

1. (a) Find \(\dfrac{dV}{dt}\). [2]
2. (b) Find the rate of change of the volume when \(t=4\). Give a reason for the sign of your answer. [2]
3. (c) Show that the volume is decreasing for \(0\le t<6\). [2]

The curve \(C\) has equation \(y=x^{3} - 2 x + 7\). The point P on \(C\) has \(x\)-coordinate \(2\). [7]

1. (a) Find the gradient of the tangent to \(C\) at P. [2]
2. (b) Find the equation of the tangent to \(C\) at P, in the form \(ax+by=c\), where \(a, b, c\in\mathbb{Z}\). [2]
3. (c) Find the equation of the normal to \(C\) at P, in the form \(y=mx+c\). [2]
4. (d) There is one other point on \(C\) at which the gradient is the same as at P. Find its \(x\)-coordinate. [1]

The curve \(C\) has equation \(y=f(x)\), where \(f(x)=x\ln(x+2)\) for \(x>0\). Give all answers to 3 significant figures. [6]

1. (a) Use your calculator to find \(f(3)\) and \(f'(3)\). [2]
2. (b) Find the equation of the tangent to \(C\) at the point where \(x=3\), in the form \(y=mx+c\). [2]
3. (c) Graph \(y=f'(x)\) and a suitable line on your calculator. Hence find the value of \(x\), for \(0<x<10\), at which the gradient of \(C\) is 1.5. [2]

The curve \(C\) has equation \(y=f(x)\), where \(f(x)=x^{4} + \frac{48}{x}\) and \(x>0\). [14]

1. (a) Find \(f'(x)\). [2]
2. (b) Show that the tangent to \(C\) at the point where \(x=2\) passes through the origin. [3]
3. (c) Find the equation of the normal to \(C\) at the point where \(x=2\). Give your answer in the form \(ax+by=c\), where \(a, b, c\in\mathbb{Z}\). [3]
4. (d) The normal meets \(C\) again at the point Q. Use your calculator to find the coordinates of Q. [3]
5. (e) The tangent to \(C\) at the point R is parallel to the line \(y=-44x+5\). Find the coordinates of R. [3]

The function \(f\) is given by \(f(x)=x^3+ax^2+bx+9\), where \(a\) and \(b\) are constants. The graph of \(y=f(x)\) has a horizontal tangent at \(x=-2\) and at \(x=4\). [14]

1. (a) Find \(f'(x)\) in terms of \(a\) and \(b\). [1]
2. (b) Hence write down two equations in \(a\) and \(b\), and find the value of \(a\) and the value of \(b\). [4]
3. (c) Write down the \(y\)-intercept of the graph. [1]
4. (d) Use your calculator to find the coordinates of the local maximum point and of the local minimum point on the graph of \(f\). [3]
5. (e) The domain of \(f\) is \(-3\le x\le 8\). Find the greatest value of \(f(x)\) on this domain. [2]
6. (f) Find the range of \(f\) on this domain. [3]

An open fish tank with a square base of side \(x\) dm and height \(h\) dm is made from 432 dm² of glass. The tank has no lid. [14]

1. (a) Show that \(h=\dfrac{432-x^2}{4x}\). [2]
2. (b) Hence show that the volume \(V\) of the tank is \(V=108x-\dfrac{x^3}{4}\). [2]
3. (c) Find \(\dfrac{dV}{dx}\). [2]
4. (d) Use your calculator to find the value of \(x\) for which \(V\) is a maximum. [3]
5. (e) Find the maximum volume and the height of the tank when its volume is a maximum. [3]
6. (f) Sketch the graph of \(V\) against \(x\) for \(0<x<20.8\), showing the maximum point. [2]