MTH109 Calculus Tutor-Marked Assignment 1, 2026 | SUSS

MTH109 Tutor-Marked Assignment 1

This assignment is worth 10% of the final mark for MTH109 Calculus.

The cut-off date for this assignment is 05 March 2026, 2355hrs.

Note to Students:

You are to include the following particulars in your submission: Course Code, Title of the TMA, SUSS PI No., Your Name, and Submission Date.

For example, ABC123 TMA01 Sally001 TanMeiMeiSally (omit D/O, S/O). Use underscore and not space.

Question 1

Determine the following limits.

Question 2

The Air Quality Health Index (AQHI) is a scale from 1 to 11 that is used to communicate the level of health risk associated with air quality. A sample of 647 readings of air pollutants taken at a monitoring station over a fixed period is used to calculate 647 values of the AQHI. Table Q2 shows the frequency counts of each value of the AQHI based on these readings. Given a sample X₁,…,X_n of n observations of the AQHI, the (uncorrected) sample variance s²(µ) relative to a parameter µ is given by

n

s²(µ) = ¹ⁿ i∑=1(X_i −µ)².

Table Q2: Frequency counts of AQHI values.

(a) Apply derivative tests by differentiating with respect to µ to show that s²(µ) is minimised when

1 n

µ = X = ⁿ i∑=1X_i ,

the sample mean.

(10 marks)

(b) Calculate the value of the sample variance s²(X) for the AQHI data in Table Q2, giving your answer correct to 2 decimal places.

(5 marks)

Question 3

(a) Define a function f : R −→ R by

where a and b are real constants.

(i) Determine the range of values of a and b such that f is continuous at x=

(5 marks)

(ii) Using the limit definition of differentiability, determine the range 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 just reaches a point 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 moves from A to C, the square of the length of the crease formed by the fold is given by the function L²(x) : [0,b] −→ R defined by

(i) Determine whether the function L²(x) is continuous over [0,b].

(5 marks)

(ii) Determine the range of values of a and b, subject to 0 < a < b < ∞, such that the function√ L²(x) has a unique maximum over [0,b] at x = b−      b² −a².

(10 marks)

Question 4

(a) Prove that the tangent line to the graph of the equation y² =x³ at the point 8/9, 16√2/27 is also a normal to the graph at some point.

(5 marks)

(b) Prove that the tangent lines to the graph of the equation y²(x−1) = x²(x+1) at the points where x = 2 intersect at an angle of .

(3 marks)

Question 5

END OF ASSIGNMENT

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