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MAT1581 Assignment 3 (COMPLETE SOLUTIONS) - Due 5 August 2025

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MAT1581 Assignment 3 (COMPLETE SOLUTIONS) - Due 5 August 2025 This assignment focuses on the application of differentiation, a fundamental concept in calculus used to analyze and understand the rate at which quantities change. The questions in this task assess the learner’s ability to determine limits, apply differentiation rules to various algebraic and trigonometric functions, and analyze the behavior of curves through concepts such as turning points and intercepts. By solving these problems, students enhance their understanding of how differentiation can be used to solve real-world mathematical problems, particularly in understanding motion, optimization, and change. The assignment aligns with the content covered in Module 6 of the MAT1581 study guide and aims to strengthen foundational calculus skills necessary for further studies in mathematics and applied sciences.

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MAT1581
Assignment 3

Due 5 August 2025

, ASSIGNMENT 03

Due date: Tuesday, 5 August 2025

Question 1: Limits (6 marks)

1.1

𝑥3 − 2 (−1)3 − 2 −3 1
𝑙𝑖𝑚𝑥→−1 4
= = =
𝑥 + 4𝑥 − 3 (−1)4 + 4(−1) − 3 6 2

1.2

𝑥3 − 8 27 − 8
𝑙𝑖𝑚𝑥→3 = = 519
𝑥2 − 4 9−4

Question 2: Differentiation (22 marks)

2.1

1
1 𝑑𝑦 𝑠𝑖𝑛𝑥 + 1
𝑦= ⇒ =− 2
𝑥 − √𝑐𝑜𝑠𝑥 𝑑𝑥 (𝑥 − √cos(𝑥) )2

Answer:

sin(𝑋) /√cos(𝑥) + 2
−
2(𝑥 − √cos (𝑥)) 2

2.2

y = (x 3 + 2)(x 2 − 2x + 1)

Using the product rule:

𝑑𝑦
= (3𝑥 2 )(𝑥 2 − 2𝑥 + 1) + (𝑥 3 + 2)(2𝑥 − 2)
𝑑𝑥

Simplified:

5x 4 − 8x 3 + 3x 2 + 4x − 4

2.3

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