Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
A complete solution with detail of all relevant formulas and skills to create in the learner a problem solving strategy.Each solution has been verified using AI. Moreover,most of the solutions have been explained graphically to create visual learning. These are not merely solutions of problems but a problem solving strategy to solve similar issues
Typology: Exams
1 / 7
This page cannot be seen from the preview
Don't miss anything!
shahbaz ahmed October 2024
Relevant formula and skills
0! = 1 nCr = (^) r!(nn−!r)! dx^ d cxn^ =^ c^ dxd xn^ =^ nxn−^1 limn−→∞(1 + (^1) n )n^ = e Where e is an irrational number such that 2 < e < 3 If ax^ = b ⇐⇒ loga b = x loge x = ln x
To expand cos(θ) using Taylor’s theorem up to three terms around x = 0, we can use the formula:
f (x) = f (0) + f ′(0)x + f^
2! x (^2) + f^ ′′′(0) 3! x (^3) + R 3 (x) where R 3 (x) is the remainder term. For cos(θ):
cos(θ) ≈ 1 + 0 · θ + −2!^1 θ^2 + 0 · θ 3 3! Simplifying this gives:
cos(θ) ≈ 1 − θ 2 2 Thus, the Taylor series expansion of cos(θ) up to three terms is:
log m + log n = log mn log mn = log m − log n log mn^ = n log m ........................................................................... Integration by parts Z u dv = uv −
v du
Q1. Find the coefficient of x^7 in the expansion of (2x − 3)^7 Circle your answer.
− 2187 − 128 2
Solution Using the formula
Tk+1 =
n k
an−kbk
Putting a = 2x, b = − 3 , n = 7
Tk+1 =
k
(2x)^7 −k(−3)k
Tk+1 =
k
27 −kx^7 −k(−3)k
Since x^7 = x^7 −k 7 = 7 − k =⇒ k = 0 Putting k=
T0+1 =
27 x^7 (−3)^0 T 1 =
27 x^7 × 1 Also 7 0
T 1 = 128x^7
Find the equation of the transformed curve. Circle your answer. To transform the curve y = ln x by a stretch parallel to the x-axis with a scale factor of 2, we replace x with x 2 :
y = ln
x 2
