Further Derivatives: exam review Exam review

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

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

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

1. (a) (a) \(y=\sqrt{9 - 2x^{2}}\) [2]
2. (b) (b) \(y=e^{\sin(3x)}\) [2]
3. (c) (c) Find the gradient of the curve \(y=(e^{x})^{3}+5x\) at the point where \(x=\ln 2\). [2]

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

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

The curve \(C\) has equation \(y=e^{2x}-5x\). [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 \(y=3x+4\). 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 function \(f\) is defined by \(f(x)=xe^{-2x}\). [8]

1. (a) (a) Find \(f'(x)\). [2]
2. (b) (b) Find the exact coordinates of the stationary point of the graph of \(y=f(x)\). [2]
3. (c) (c) Find \(f''(x)\). [2]
4. (d) (d) Use your answer to part (c) to show that the stationary point is a maximum, and find the \(x\)-coordinate of the point of inflexion. [2]

The concentration, \(C\) ppm, of a pollutant at a distance of \(x\) km from a factory is modelled by \(C=\frac{160}{x^{2}}\). A drone flies directly away from the factory at a constant speed of 0.4 km per minute. [5]

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

A particle moves in a straight line. The displacement from a fixed point O is \(s\) metres after \(t\) seconds, where \(s=2 t^{3} - 15 t^{2} + 24 t\) 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 \(t=3\). Justify your answer. [3]

A bead slides along a straight wire. The displacement from a fixed point O is \(s\) metres after \(t\) seconds, where \(s=8\left(1-e^{-0.5t}\right)\) for \(t\ge0\). [6]

1. (a) (a) Find an expression for the velocity. [2]
2. (b) (b) Find the time when the velocity is \(1.5\) m s\(^{-1}\). [2]
3. (c) (c) Find the distance from O that the particle approaches as \(t\) becomes very large. [2]

A funnel is in the shape of an inverted cone of height 12 cm and radius 4 cm at the top. Liquid flows into the funnel at a constant rate of 5 cm³ per second. At time \(t\) seconds the depth of the liquid is \(h\) cm and the volume of liquid is \(V\) cm³. [12]

1. (a) (a) Show that \(V=\frac{\pi h^{3}}{27}\). [2]
2. (b) (b) Find the rate at which the depth is increasing at the instant when \(h=6\). [4]
3. (c) (c) Find the rate at which the area of the liquid surface is increasing at the instant when \(h=6\). [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 \(f(x)=x^{3}+ax^{2}+bx+5\), where \(a\) and \(b\) are constants. The graph of \(y=f(x)\) has a point of inflexion at \(x=1\) and a stationary point at \(x=3\). [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 \(x=1\). [2]
4. (d) (d) Find the set of values of \(k\) for which \(f(x)=k\) has exactly three solutions. [2]

A drinks company makes closed cylindrical cans of volume 330 cm³. A can 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 \(A=2\pi r^{2}+\frac{660}{r}\). [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 deer, \(P\), on an island \(t\) years after they are first counted is modelled by \(P(t)=\frac{1200}{1+11e^{-0.4t}}\), \(t\geq 0\). [13]

1. (a) (a) Find the initial population 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 population is changing when \(t=2\). [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 robot arm slides a component along a straight rail. Its velocity at time \(t\) seconds is \(v\) m s\(^{-1}\), where \(v=3\cos(2t)\) for \(t\ge0\). The displacement of the component from a fixed point O is \(s\) metres, and \(s=1\) 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 component is momentarily at rest. [2]
3. (c) (c) Write down the greatest speed of the component. [1]
4. (d) (d) Find an expression for \(s\) in terms of \(t\). [2]
5. (e) (e) Find the displacement of the component from its starting position when \(t=4\). [2]
6. (f) (f) Find the total distance travelled by the component in the first 4 seconds. [3]