SIT190 Assignment 3 Due Date: Friday, 21st September, 2018, 5.00 pm Complete the following problems, showing all your working. Marks are allocated to your steps, not just the final answer. 1. Find ??...

i attached my assignment might need working out aswell


SIT190 Assignment 3 Due Date: Friday, 21st September, 2018, 5.00 pm Complete the following problems, showing all your working. Marks are allocated to your steps, not just the final answer. 1. Find ?? ?? for: (i) ? = −3ln ?2 (ii) ? = ?7? (iii) ? = 3 sin(?) − 1 2 cos(2?) (iv) ? = − 8 sin ( ? 4 ) + 6cos ( 2? 3 ) (v) ? = 4 ?3 + 1 8 ?−8? 2. It takes 200 minutes to fill a water tank. The volume of water ? litres, after time ? minutes, is given by ? = 100? − 1 4 ?2, for 0 ≤ ? ≤ 200 (i) Find the instantaneous rate of change of volume when ? = 100. (ii) Find the volume of water in the tank when the instantaneous rate of change of volume is 20 litres/minute. (iii) Find the instantaneous rate of change of volume when the volume of water in the tank is 6400 litres. (iv) Will the tank overflow? Give a reason for your answer. 3. For each of the following functions find the ? and ? co-ordinates of the stationary points. Then classify each of the stationary points as a local maximum, a local minimum, or a horizontal point of inflection (using the First Derivative Test). (i) ? = 4?2 − 32? + 14 (ii) ? = 1 3 ?3 − 3?2 + 9? + 8 (iii) ? = 1 3 ?3 + 1 2 ?2 − 12? − 4 4. For ? = 1 5 ?5 − 3?3, (i) find the ? and ?-intercepts (ii) find and classify the stationary points (using the First Derivative Test) (iii) sketch, labelling all intercepts and stationary points. 5. Find ?? ?? for: (i) ? = − 1 4 ?4 cos ? (ii) ? = 4−6? 1 2 ?2−3 (iii) ? = 2?2−4 sin( −2?) (iv) ? = (3?4 − 2)?? (v) ? = cos(2?) + ?5ln (?) 6. Use the chain rule to find ?? ?? for: (i) ? = (− 1 3 ?3 + 5 2 ?2 + 3?)4 (ii) ? = ln (sin( 2?)) (iii) ? = ? ( − ?5 5 ) (iv) ? = ?(sin(?)−?) 7. Find ?? ?? and ?2? ??2 for: (i) ? = 1 8 ?8 + 3?2 − 7? (ii) ? = ?2?3? 8. Given that ? = 5 3 ?3 + 10?2 − 6, (i) Find all stationary points (ii) Use the second derivative test to classify the points found in (i). 9. Find: (i) ? = ∫(8?3 − 9?2 + 8? − 2) ?? (ii) ? = ∫ ?−4(6?6 + 4?5) ?? (iii) ? = ∫(4? − 3 2 − 3 2 √? − 7 2√? ) ?? (iv) ? = ∫(4?4? + 3?−? + 5 ? ) ?? (v) ? = ∫(−3 sin(3?) + 6 cos(6?)) ?? 10. Find ? if (i) ?? ?? = 6?2 + 4? − 10 and ?(1) = 2 (ii) ?? ?? = 6 ? + 9?2 − 12? and ?(1) = 4 11. The velocity of an object is given by ? = 9?2 − 6? (m/sec) for 0 ≤ ? ≤ 5. If the object is initially at rest, find: (i) the velocity after 2 seconds (ii) the acceleration after 2 seconds (iii) the distance travelled after 5 seconds (iv) the velocity when the acceleration is 48 m/sec2 12. Evaluate: (i) ? = ∫ (6?2 − 4? + 3) ?? 4 3 (ii) ? = ∫ (7√?5 − 1) ?? 1 0 (iii) ? = ∫ cos(2?) ?? 5? 4 3? 4 13. For ? = ?2 + 2? − 3, (i) Sketch the curve for 0 ≤ ? ≤ 2 marking on your sketch the ? and ? intercepts and ?(2). (ii) Find the area enclosed between the curve and the ? axis for 0 ≤ ? ≤ 2