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Prepared for internal exercise only! 1. Evaluate the following: x²-y6 lim a. (х.у)- (0,0) хуз дт b. Let z = e¯* sin(y), where x = s t² and y = s² t. Find using chain rule ds c. The ideal gas law'for a given gas is PV = 8.31T, where P is the pressure (in kilopascals), V is the volume (in liters), and T is the temperature (in kelvins). Find the rate at which the pressure is changing when the temperature is 300 K and increasing at a rate of 0.1 K/s. The volume is 100 L and increasing at a rate of 0.2 L/s. Let f(x, y) = x²y. Find V(3,2) and find the derivative of f in the direction of (1,2) at the point (3,2). d. e. Find the volume under the surface z = V1 – x² and above the triangle formed by y = x, x = 1, and the x-axis. A lamina has shape of the region in the first quadrant that is bounded by the graphs of y = sin x, y = cos x, between x = 0 and x = n/4. Determine the centre of mass if the density is y. a. b. A cylindrical tank is 1 m high and 0.3 m radius. If height is increased by 5 cm and radius by 1 cm what is the effect on volume? c. The angle of elevation of the top of a tower is found to be 30° + 0.5° from a point300 + 0.1 m from the base. Estimate the towers height. d. 2.