MTH107 Determine limit (t- > 2) t ^ 3 – 8 / t ^ 2 – 4: Calculus I Tutor Marked Assignment, Singapore

MTH107 Calculus I Tutor-Marked Assignment

Question 1

(a) Determine limit (t- > 2) t ^ 3 – 8 / t ^ 2 – 4.

(b) The inequality 1 – (x^2 / 6)< sin x / x < 1 holds for x is measured in radians and |x| < 1. Use the inequality to find the limit(x->0) sin x / x.

(c) Evaluate limit(x->0) x (e^x – 1) / 1 – cos2x.

Question 2

(a) Evaluate limit(x->0) x*e^x-log(1+x)/x^2.

(b) Find the values of the constants a and b such that limit(x->0) x(1+a*cosx)-b*sinx/x^2=1.

Question 3

(a) Consider the y = f(x) defined by f(x) = { x* sin(1/x) if x is not = 0; 0 if x is = 0

Is the function continuous at x = 0? Explain your answer.

(b). Find the value of the constant a to make the function f(x)= { x^2 – 1     x < 3; 2ax   x>=3 continuous.

(c). Consider the function f(x) = { x    0<=x<1; 2 – x     1<x<=2. Does f have a derivative at x =1? Explain your answer.

Question 4

(a) Explain with reasons why the equation x^3+3x+1=0 has exactly one real root.

(b) Demonstrate that for any two real numbers a, b ∈ R |sinb –  sina| <= |b-a|.

(c) State the conditions for the Rolle’s theorem and verify that the conditions are satisfied for the function f = sin x / e^x (in 0, pi).

Hence verify the conclusion of the theorem.

Question 5

(a) Find the Taylor series expansion of the function loge x about the point a =1. Use the series to evaluate the limit, limit x->1 ln x/ x-1.

(b) A box with a square base and open top are to hold 32 cube centimetre of a liquid. Find the dimensions that require the least amount of material, neglecting the thickness of the material, of the material and the waste in the construction.