University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

452

Pages:

3
Author:

MAlphen

Differentiation (Maxima and Minima) A piece of wire of length 180 cm is bent into the shape 𝑃𝑄𝑅𝑆𝑇 shown in the diagram. Show that the area, 𝐴 cm2, enclosed by the wire is given by 𝐴 = 2160 − 540𝑥2. Find the value of 𝑥 and 𝑦 for which 𝐴 is a maximum. Ans: Differentiation and Integration 1. (i) Find 𝑑 [𝑒2𝑥 (cos 3𝑥 + 𝑑𝑥 3 2 sin 3𝑥)]. (ii) Hence find ∫ 𝑒2𝑥 cos 3𝑥 𝑑𝑥. 3 Ans: (i) 13 𝑒2𝑥𝑐𝑜𝑠3𝑥, (ii) 2 𝑒2𝑥 (𝑐𝑜𝑠3𝑥 + 𝑠𝑖𝑛3𝑥) + 𝐶 2 1. Express 13 2𝑥−1 𝑥2(𝑥+1) in partial fractions. (ii) Hence, determine ∫ Ans: (i) 3 − 𝑥 1 − 𝑥2 2 3 2𝑥−1 𝑥2(𝑥+1) 𝑑𝑥. (ii) 𝑥+1 1 3𝑙𝑛𝑥 + 𝑥 ( − 3 ln 𝑥 + 1 ) +𝐶