Using maclaurin’s theorem, and using Application of differentiation

Question

Question 1. Using maclaurin’s theorem expand sinx upto the term x^7.

Question 2.

(a) 1000m of fencing is to be used to make a rectangular enclosure. find the greatest possible area and the        corresponding dimensions.

 (b) use differentials to approximate (square root) 65.

(3) Given the curve y=4x³-3x^4.

  (i) state the stationary (turning) points of the curve.

  (ii) sketch the curve.

Answer 1

Question 1.

sin(x) = x - (X³)/3! +(x⁵)/5! - (x⁷)/7!

 

Question 2a.

A = lw Lets say the length is x. We ha 2 lengths so we would have used 2x of fence. Since we have 2 widths the width of one side would be (1000 - 2x)/2. So,

A = x(1000 - 2x)/2

A = 1000x/2 - 2x²/2

A = 500x - x² Now take that derivative and set it equal to 0 to find the max.

A' = 500 - 2x

500 - 2x = 0

x = 250 Plug this in to your length and width

l = 250 w = 250 it's a square

 

Question 2b.

f(x) = Sqrt(x)

Let a = 64, b = 65, and Δx = b - a = 1

We want f(a) + f'(a)(Δx)

f(a) = Sqrt(64) = 8

f'(x) = 1/(2Sqrt(x))

f'(a) = 1/(2Sqrt(64)) = 1/16

f(a) + f'(a)(Δx) = 8 + 1/16(1) = 129/16