MYTUTOR SUBJECT ANSWERS

560 views

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Let f be a function defined on the positive real axis with values in R, such that f(x) = x - ln(1+x). Differentiating this function, we obtain, f'(x) = (x - ln(1+x))' = x' - ln'(1+x) = 1 - 1/(1+x). Since, x > 0, we have 1 + x > 1, and so 1/(1+x) < 1. So, 1 - 1/(1+ x) > 0. So, f'(x) > 0, for all positive x. So, by L'Hospital Rule, f(x) is strictly increasing. Thus, f(x) > lim f(x) when x -> 0 = 0 - ln (1+0) = 0. So, f(x) > 0, for all positive. x. 

Andreea I. Uni Admissions Test .MAT. tutor, Uni Admissions Test .STEP...

1 year ago

Answered by Andreea, an Uni Admissions Test .STEP. tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

17 SUBJECT SPECIALISTS

Ross G. GCSE Maths tutor, A Level Maths tutor, GCSE Further Mathemati...
£28 /hr

Ross G.

Degree: Mathematics (Masters) - Durham University

Subjects offered:.STEP., Physics+ 2 more

.STEP.
Physics
Maths
Further Mathematics

“I study maths, the subject I love, at Durham University. I hope I can help you to learn the content and, maybe, you can learn to love maths too!”

£25 /hr

Guy P.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:.STEP., Maths+ 2 more

.STEP.
Maths
Further Mathematics
.MAT.

“Hi there. Thank you for taking the time to visit my profile, it would be great to meet in a FREE 15-minute meeting! I look forward to hearing from you soon!”

£30 /hr

Henri F.

Degree: Aerospace Engineering MEng (Masters) - Bristol University

Subjects offered:.STEP., Physics+ 4 more

.STEP.
Physics
Maths
Further Mathematics
Electronics
.PAT.

“Aerospace Engineering PhD candidate in spacecraft control with 6 years of experience in tutoring. ”

About the author

£25 /hr

Andreea I.

Degree: Mathematics and Computer Science (Masters) - Oxford, Merton College University

Subjects offered:.STEP., Maths+ 1 more

.STEP.
Maths
.MAT.

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Posts by Andreea

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

How many distinct real roots does the equation x^3 − 30x^2 + 108x − 104 = 0 have?

The quadratic equation 2x^2 + 8x + 1 = 0 has roots a and b. Write down the value of a + b, a*b and a^2 + b^2.

Other Uni Admissions Test .STEP. questions

Show that substituting y = xv, where v is a function of x, in the differential equation "xy(dy/dx) + y^2 − 2x^2 = 0" (with x is not equal to 0) leads to the differential equation "xv(dv/dx) + 2v^2 − 2 = 0"

How can I integrate e^x sin(x)?

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Prove that any number of the form pq, where p and q are prime numbers greater than 2, can be written as the difference of two squares in exactly two distinct ways.

View Uni Admissions Test .STEP. tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok