Calculus Problems

1. (a) A lamina is defined by the regions R1 and R2: Using a well proportioned and neatly drawn graph of these regions, identify the area corresponding to the lamina. (b) Using integral calculus, calculate the area defined in part a) of this question. (c) Using integral calculus determine the volume obtained if the area identified in part a) of this question is swept 360° around the positive x-axis. 2. (a) Using the complex numbers, 21 = 3 - 4i, and to=2+3i,calcu- late; i. 21 + 22 ii. 21 - 22 iii. (b) Using Euler's formula, 20 = cos () + isin(8), derive expressions in terms of complex exponentials for; i. cos(8) ii. sin(0) iii. tan (0) iv. Ecos () (c) The results of an experiment allowed the following equations to be derived; 54 i. Write the system of equations in the matrix form, A.X = B, and calculate the determinant of the A matrix. ii. Calculate the matrix of co-factors of the A matrix. iii. Calculate the inverse matrix for the A matrix, and demon- strate that the answer is the inverse of A. iv. Using the inverse matrix, A-! find the values of the unknown values, a, 3, 3. (a) On a dual carriageway, the front edge of a car overtaking a truck draws level with the front edge of the truck. Exactly at this time, 70m. ahead, a large tree falls across both carriageways. The truck is travelling at a constant 90k-ph, and the car, a con- stant 110kph. The driver of the truck reacts quickly, and the truck's brakes are fully applied 0. 5s after the tree falls, thereafter, giving a constant deceleration of the truck of 0.6g. The driver of the car was distracted, and the car's brakes were not fully applied until 0.7s after the tree fell, thereafter, giving a constant deceleration of the car of 0.8g. Assume that the acceleration due to gravity, g, is 9.81m/s². (i) Calculate the distance travelled by each vehicle, assuming constant velocity, before their brakes were applied. This is the so-called "thinking distance". (ii) Calculate the total stopping distance for the truck (thinking distance + stopping distance). (iii) Calculate the velocity with which the car hits the tree. (b) The following data were obtained from an experimental test. Find the linear regression line of the data and determine the linearity coefficient of the regression line. X 1 2 3 4 5 6 7 8 9 10 Y 20 16 12 9 5 -1 -7 -7 -13 -20 -15 (c) Find the general solution for the following first order ordinary differential equation dy =2y'sin(x) = 4. (a) A water tank is shown in figure 1. The tank has diameter, D = 2.4 m and a hole of diameter, d = 4 cm in the centre of the bottom of the tank. The initial depth of water in the tank is, no = 2 m. Assume that the water velocity obeys Torricelli's Law, U = V 2gh, where g = 9.81 m/s2 (acceleration due to gravity). D h. o d Figure 1: Water Tank Calculate the time for the tank to empty to one quarter its initial level. (b) Find the particular solution for the following ordinary differential equation which has the boundary condition, I = 0, when y = 3. 5. (a) The volume of a solid, V, bounded by the curve 2 = 4.2 + 3y - 6 between the limits T = 0 and x = 4cm, and y = 0 and y = 6cm is given by; V = lo 0 4 / 6 4r² + 3y - 6 dy dr Evaluate the volume. (b) (i) Find the general solution for the following ordinary differen- tial equation = - 5 dx 3y2 + 5y + 4 (ii) Find the particular solution to the above ordinary differen- - tial equation with the initial conditions, when T = 0, y = 3. 6. An inductor, capacitor, resistor series circuit is shown in figure 2. L represents the inductance (H), C, the capacitance (F), R, the re- sistance (), i, the current (A), and V, the potential difference (V). Assume that each component is "ideal". For example, that the in- ductor and capacitor have no dissipation. /YYYYYY\ L V(t) c R i Figure 2: Inductor, Capacitor, Resistor, Series Circuit (a) Derive the differential equation for the circuit in terms of the charge, q. Note, q = J idt. (b) If the inductance, L, is 0.02H, the resistance, R, is 2209, the capacitance, C, 3x10-GF, and the applied voltage, V, 5e-300 V. Find the General Solution for the circuit. (c) Determine the Particular Solution if g(0) = 0, and = 10C/s