Further Derivatives: exam review Exam review: second paper

Differentiate each function with respect to \(x\). Simplify your answers where you can. [6]

1. (a) (a) \(f(x)=\frac{10x^{4}-6x^{2}}{\sqrt{x}}\) [3]
2. (b) (b) \(g(x)=3\ln\left(x^{7}\right)+5\tan x\) [3]

Differentiate with respect to \(x\). [6]

1. (a) (a) \(y=\sqrt[3]{5x^{2} + 7}\) [2]
2. (b) (b) \(y=e^{\cos(2x)}\) [2]
3. (c) (c) Find the gradient of the curve \(y=(e^{x})^{2}+7x\) at the point where \(x=\ln 5\). [2]

Differentiate with respect to \(x\). Show the rule you use. [6]

1. (a) (a) \(y=x^{3}\ln(2x-1)\) [3]
2. (b) (b) \(y=\frac{e^{x}}{(3x-2)^{2}}\) [3]

The curve \(C\) has equation \(y=e^{3x}-7x\). [7]

1. (a) (a) Find \(\frac{dy}{dx}\). [1]
2. (b) (b) The tangent to \(C\) at the point P is parallel to the line \(y=5x-1\). Find the exact \(x\)-coordinate of P. [3]
3. (c) (c) Find the equation of the normal to \(C\) at the point where \(x=0\). Give your answer in the form \(y=mx+c\). [3]

The concentration of a drug in a patient's blood, in mg per litre, is modelled by \(f(t)=10te^{-0.5t}\), where \(t\) is the time in hours after the dose is taken, \(t\geq 0\). [8]

1. (a) (a) Find \(f'(t)\). [2]
2. (b) (b) Find the exact coordinates of the stationary point of the graph of \(y=f(t)\). [2]
3. (c) (c) Find \(f''(t)\). [2]
4. (d) (d) Use your answer to part (c) to show that the stationary point is a maximum, and find the \(t\)-coordinate of the point of inflexion. [2]

The light intensity, \(I\) lux, at a distance of \(x\) metres from a lamp is modelled by \(I=\frac{250}{x^{2}}\). A moth flies directly away from the lamp at a constant speed of 0.6 metres per second. [5]

1. (a) (a) Find \(\frac{dI}{dx}\). [2]
2. (b) (b) Find the rate of change of \(I\) when \(x=5\). [2]
3. (c) (c) Explain what the sign of your answer to part (b) tells you. [1]

A cyclist rides along a straight road. The displacement from a fixed point O is \(s\) metres after \(t\) seconds, where \(s=2 t^{3} - 21 t^{2} + 60 t\) for \(t\ge0\). [7]

1. (a) (a) Find expressions for the velocity and the acceleration. [2]
2. (b) (b) Find the times when the particle is at rest. [2]
3. (c) (c) Determine whether the particle is speeding up or slowing down when \(t=6\). Justify your answer. [3]

A toy train runs along a straight track. The displacement from a fixed point O is \(s\) metres after \(t\) seconds, where \(s=12\left(1-e^{-0.25t}\right)\) for \(t\ge0\). [6]

1. (a) (a) Find an expression for the velocity. [2]
2. (b) (b) Find the time when the velocity is \(0.9\) m s\(^{-1}\). [2]
3. (c) (c) Find the distance from O that the particle approaches as \(t\) becomes very large. [2]

A storage tank is in the shape of an inverted cone of height 15 m and radius 6 m at the top. Water is pumped into the tank at a constant rate of 8 m³ per minute. At time \(t\) minutes the depth of the water is \(h\) m and the volume of water is \(V\) m³. [12]

1. (a) (a) Show that \(V=\frac{4 \pi h^{3}}{75}\). [2]
2. (b) (b) Find the rate at which the depth is increasing at the instant when \(h=10\). [4]
3. (c) (c) Find the rate at which the area of the liquid surface is increasing at the instant when \(h=10\). [4]
4. (d) (d) Explain why the depth rises more and more slowly while the liquid flows in at a constant rate. [2]

The function \(f\) is defined by \(f(x)=x^{3}+ax^{2}+bx+10\), where \(a\) and \(b\) are constants. The graph of \(y=f(x)\) has a point of inflexion at \(x=2\) and a stationary point at \(x=-1\). [13]

1. (a) (a) Find the value of \(a\) and the value of \(b\). [4]
2. (b) (b) Find the coordinates of the stationary points of the graph of \(y=f(x)\). Use the second derivative to decide the nature of each. [5]
3. (c) (c) Write down the set of values of \(x\) for which the graph is concave down. Explain why there is a point of inflexion at \(x=2\). [2]
4. (d) (d) Find the set of values of \(k\) for which \(f(x)=k\) has exactly three solutions. [2]

A paint company makes closed cylindrical tins of volume 1000 cm³. A tin has base radius \(r\) cm and height \(h\) cm. The company wants to use as little metal as possible. [12]

1. (a) (a) Write down an expression for \(h\) in terms of \(r\). [1]
2. (b) (b) Show that the total surface area is \(A=2\pi r^{2}+\frac{2000}{r}\). [2]
3. (c) (c) Find \(\frac{dA}{dr}\). [2]
4. (d) (d) Find the value of \(r\) that minimizes the surface area. Give your answer to 3 significant figures. [3]
5. (e) (e) Use the second derivative to justify that this value of \(r\) gives a minimum. [2]
6. (f) (f) Find the minimum surface area. [2]

The number of members, \(P\), of an online chess club \(t\) months after it opens is modelled by \(P(t)=\frac{4500}{1+14e^{-0.7t}}\), \(t\geq 0\). [13]

1. (a) (a) Find the initial membership at \(t=0\). [2]
2. (b) (b) Write down the value that \(P\) approaches as \(t\to\infty\). [1]
3. (c) (c) Find \(\frac{dP}{dt}\). [3]
4. (d) (d) Find the rate at which the membership is changing when \(t=3\). [2]
5. (e) (e) Use your calculator to find the greatest rate of change of \(P\) and the value of \(t\) at which it occurs. [3]
6. (f) (f) Find the value of \(P\) at this time and describe what happens to the graph of \(P\) at this point. [2]

A piston head moves in a straight line inside a cylinder. Its velocity at time \(t\) seconds is \(v\) m s\(^{-1}\), where \(v=4\sin(1.5t)\) for \(t\ge0\). The displacement of the piston head from a fixed point O is \(s\) metres, and \(s=2\) when \(t=0\). [12]

1. (a) (a) Find an expression for the acceleration. [2]
2. (b) (b) Find the first time \(t>0\) when the piston head is momentarily at rest. [2]
3. (c) (c) Write down the greatest speed of the piston head. [1]
4. (d) (d) Find an expression for \(s\) in terms of \(t\). [2]
5. (e) (e) Find the displacement of the piston head from its starting position when \(t=5\). [2]
6. (f) (f) Find the total distance travelled by the piston head in the first 5 seconds. [3]