MTH109 Tutor-Marked Task 1 This assignment is fee 10% of the last sign for MTH109 Calculus. The closing date for this assignment is 05 March 2026, 2355hrs. Veil to Students: You are to incorporate the next particulars to your submission: Path Code, Title of the TMA, SUSS PI No., Your Title, and Submission Date. As an illustration, ABC123 TMA01 Sally001 TanMeiMeiSally (omit D/O, S/O). Exhaust underscore and never plan. Question 1 Resolve the next limits. Question 2 The Air Quality Successfully being Index (AQHI) is a scale from 1 to 11 that is outdated to communicate the stage of successfully being chance linked to air quality. A pattern of 647 readings of air pollution taken at a monitoring residing over a assign duration is outdated to calculate 647 values of the AQHI. Desk Q2 shows the frequency counts of every fee of the AQHI in step with these readings. Given a pattern X 1 ,…,Xn of n observations of the AQHI, the (uncorrected) pattern variance s 2(µ) relative to a parameter µ is given by n s 2(µ) = 1n i∑=1(Xi −µ)2 . AQHI 1 2 3 4 5 6 7 8 9 10 11 Frequency 0 38 197 206 152 47 7 0 0 0 0 Desk Q2: Frequency counts of AQHI values. (a) Apply derivative tests by differentiating with appreciate to µ to prove that s 2(µ) is minimised when 1 n µ = X = n i∑=1Xi , the pattern mean. (10 marks) (b) Calculate the cost of the pattern variance s 2(X) for the AQHI info in Desk Q2, giving your resolution moral to 2 decimal locations. (5 marks) Question 3 (a) Define a perform f : R −→ R by where a and b are precise constants. (i) Resolve the fluctuate of values of a and b such that f is staunch at x= (5 marks) (ii) The use of the restrict definition of differentiability, resolve the fluctuate of values of a and b such that f is differentiable at x = (10 marks) (b) A rectangular sheet of paper OACB is folded over so that the corner O actual reaches a degree P on the side AC as shown in Figure Q3b. Let OA = a and OB = b, where 0 < a < b, and let x = AP. As P strikes from A to C, the square of the dimensions of the crease fashioned by the fold is given by the perform L 2(x) : [0,b] −→ R outlined by (i) Resolve whether the perform L 2(x) is staunch over [0,b]. (5 marks) (ii) Resolve the fluctuate of values of a and b, enviornment to 0 < a < b < ∞, such that the perform√ L 2(x) has a wander most over [0,b] at x = b− b 2 −a 2. (10 marks) Question 4 (a) Trace that the tangent line to the graph of the equation y 2 =x 3 on the point 8/9, 16√2/27 is additionally a protracted-established to the graph in some unspecified time in the future. (5 marks) (b) Trace that the tangent lines to the graph of the equation y 2(x−1) = x 2(x+1) on the capabilities where x = 2 intersect at an perspective of . (3 marks) Question 5 END OF ASSIGNMENT
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