# Does this model accurately reflect how water flows out of a reservoir?

From first principles, find the derivative of the function f(x) = 4x 3 − x 2 − 5. That is, find lim h→0 f(x + h) − f(x) h . [10 marks] 2. Using the rules of differentiation (that is, not from first principles) find the derivatives of each of the following functions. If you use the product rule, quotient rule or chain rule, be sure to indicate where you have done that. Do not simplify your answers. (a) f(x) = tan x 2 + 2x. (b) f(x) = (x 2 + 1)(x 3 − 1). (c) g(x) = x 3 − 1 x 2 + 1 . (d) h(x) = r sin x 1 + cos x . [4 + 4 + 4 + 4 = 16 marks] 3. Find the domain and range for each of the functions f, g, and h below, justifying your answers. Use radians for trigonometric functions. (a) f(x) = 1 1 − x 2 , (b) g(x) = cos x, (c) h(x) = f(g(x)). [3 + 2 + 4 = 9 marks] 4. (a) Verify that the line y = −2x − 1 goes through the point (1, −3) and that it is a tangent to the graph of y = x 2 . Hint: There should be precisely one point common to the line and the parabola, so precisely one solution to −2x − 1 = x 2 . There is another line through point (1, −3) that is a tangent to y = x 2 . Find its equation, and the coordinates of the point where it touches the parabola. [4 + 5 = 9 marks] 2 5. A civil engineering student wants to understand how water flows out of a reservoir through a drain hole at its bottom. She decides to use calculus to build a model of the water flow. If V (t) represents the volume of water inside the reservoir at time t, she thinks it is reasonable to say that the more water there is in the reservoir, the faster the water will flow out. The quantity dV dt represents the rate at which the volume of water in the reservoir changes. At time t = 0 the reservoir contains V (0) = 500 litres of water. After some thought, the student settles on the model dV dt = −kV (t), where k > 0 is a fixed (but unknown) constant. (a) Why did she put a negative sign in her equation? Justify. (b) After doing some theoretical work, she concludes that V (t) = 500e −kt is the solution to the model. Verify that she is correct. (c) How could the value of k be determined? (d) Criticise the model: In your opinion, does this model accurately reflect how water flows out of a reservoir? Justify. [2 + 3 + 3 + 3 = 11 marks] 6. A popular brand of chocolate (see https://en.wikipedia.org/wiki/Toblerone) is sold in the form of a long bar with a triangular cross-section. You are asked to design a box for this chocolate bar. The cross-section of the box is to be an equilateral triangle. The surface area of the box is to be 500 cm2 . The volume is to be as large as possible. Model this problem, but don’t carry out the calculations needed to solve it. You may need to use some simple geometry. Your answer should include (a) a labelled sketch of the box, (b) statements defining appropriate variables (with units), (c) a formula for a function that would have to be maximised or minimised, and (d) equations that encode any relationships between your variables. [3 + 3 + 2 + 4 = 12 marks] [Total 67 marks]