MATH1013: Mathematics and Applications 1 Assignment | Australian National University

MATH1013: Mathematics and Applications 1 Assignment 2

Learning Outcomes:

Upon successful completion, students will have the knowledge and skills to:

• Explain the fundamental concepts of calculus and linear algebra and their role in modern mathematics and applied contexts.  These concepts include the solution of linear systems, matrix algebra, linear transformations and determinants in Linear Algebra; and limits, continuity, differentiation, local and absolute extrema, Riemann integration and the fundamental theorem of calculus in Calculus.
• Demonstrate accurate and efficient use of calculus and linear algebra techniques as they relate to the concepts listed above.
• Demonstrate capacity for mathematical reasoning through explaining concepts from calculus and linear algebra and reproducing familiar proofs.
• Apply problem-solving using calculus and linear algebra techniques applied to diverse situations in physics, engineering and other mathematical contexts.

1   Calculus Question 1 (10 marks)

In the course we have discussed the following techniques to evaluate definite integrals: evaluate a limit of Riemann Sums; use area formulas for familiar shapes; use the Fundamental Theorem of Calculus II; approximate value using numerical integration.

Using the FTC II requires finding antiderivatives. We have discussed the following techniques for antidifferentiation: recognising the integrand as the derivative of a familiar function; algebraic manipulation of the integrand to represent the integrand in a different way to facilitate anti differentiation; substitution; integration-by-parts; inverse-trig substitution; the method of partial fractions.

a) In no more than one page, write a guide for how to decide which method(s) to try when faced with an integration problem. Your guide may be in the form of a flow-chart, a bulleted list, or text.

b) Demonstrate the effectiveness of your guide by evaluating the following. For each, you should explain how your guide helped you to use the method you used. You may use an online partial fractions decomposition calculator (such as https://www.wolframalpha.com/calculators/partial-fraction-calculator) if necessary.

(i) ∫ (x² + 8x − 3)/(x³ + 3x²) dx

(ii) ∫ √(x² − 1) dx

(iii) ∫ x ln x dx

(iv) ∫ (2√x)/√x dx

2  Calculus Question 2 (6 marks)

Let f(x) = e^(x²) and let I = ∫₀³ f(x) dx. Since there is no elementary antiderivative of f(x), we cannot use the FTC II to evaluate I. Instead, we may use the Trapezoidal Rule to approximate I to any required degree of accuracy.

a) Compute f′(x), f″(x) and f‴(x).

b) Use the Closed Interval Method to find the maximum and minimum values taken by f″(x) over the interval [0,3].

c) Use the Error Bound for the Trapezoidal Rule to determine the minimum value of n that will ensure that |E_T| < 0.001.

d) Evaluate T correct to 3 decimal places, where n is the value you determined in (c).
(You may use a spreadsheet or a calculator to perform this calculation, but you must provide something that demonstrates all of your working.)