Study Guides (248,280)
Canada (121,450)
Mathematics (105)
Quiz

Assignment 6 Solutions.pdf

6 Pages
114 Views
Unlock Document

Department
Mathematics
Course
Mathematics 0110A/B
Professor
Chris Brandl
Semester
Fall

Description
MATH 127 Fall 2012 Assignment 6 Solutions 1. Compute the indicated derivatives for the following functions. 2 6x ▯ x 0 (a) f(x) = 2 p , f (x) x + x ▯ ▯ ▯ 2 2 p 2▯ 1 6 + x2)(x + x) ▯ (6x ▯ x 2x + 2 x f (x) = p (x + x)2 ▯ p ▯ ▯ p ▯ 6x + 2 + 6 x + 2 ▯ 12x ▯ 4 + 3 x ▯ 1 x2 x2 = 2 p 2 (x + x) ▯6x + 3 px + 3 + 6 x2 = 2 p 2 (x + x) 2 x 2 00 (b) f(x) = ▯ 2, f (x) 2 x 0 4 f (x) = x + 3 x 00 12 f (x) = 1 ▯ 4 x x +1 2 0 (c) f(x) = e ln(x + 1), f (x) ▯ ▯ f (x) = (2x)ex +1 ln(x + 1) + ex +1 2x x + 1 cos(▯ x) (d) f(x) = , f (x) sin(x) p p ▯ p sin(▯ x)(sin(x)) ▯ cos(▯ x)cos(x) f (x) = 2 x (sin(x))2 2. Find f (x) ▯ 3 (a) f(x) = x arctan(sin(x )) 2 3 f (x) = ▯x ▯▯1arctan(sin(x )) + x▯ 3x cos(x ) 1 + (sin(x )) 1 ▯ ▯ 1 ln(x) (b) f(x) = x We will use logarithmic di▯erentiation to ▯nd the indicated derivative. We ▯rst take the logarithm of both sides of the equation and simplify. ▯ ▯ 1 ln(x) y = x ▯ ▯ ln(x) 1 ln(y) = ln x ▯ 1 = ln(x)ln x = ▯(ln(x))2 Di▯erentiating both sides we get y0 2ln(x) = ▯ y x 0 2ln(x) y = ▯ y x ▯ ▯ 2ln(x) 1 ln(x) = ▯ x x Thus, we have ▯ ▯ ln(x) 0 2ln(x) 1 f (x) = ▯ x x x (c) f(x) = (cos(x)) We will compute the derivative of (cos(x)) using logarithmic di▯erentiation. y = (cos(x)) x ln(y) = ln((cos(x)) ) = xln(cos(x)) ▯ ▯ y0 sin(x) = ln(cos(x)) + x ▯ y cos(x) 0 y = (ln(cos(x)) ▯ xtan(x))y = (ln(cos(x)) ▯ xtan(x))(cos(x))x 2 Thus, we have 0 x f (x) = (ln(cos(x)) ▯ xtan(x))(cos(x)) p arcsin(x) cos(x) + 2 (d) f(x) = x 3 , ▯nd the derivative without using the product or e (x + x) quotient rules. p arcsin(x) cos(x) + 2 We will use logarithmic di▯erentiation on y = e (x + x) . We ▯rst take the logarithm of both sides and simplify. p ! arcsin(x) cos(x) + 2 lny = ln 2 e (x + x) ▯ ▯ ▯ ▯ p x 3 = ln arcsin(x) cos(x) + 2 ▯ ln e (x + x) ▯p ▯ ▯ ▯ ▯ ▯ = ln(arcsin(x)) + ln cos(x) + 2 ▯ ln e x2 ▯ ln x + x ▯ ▯ = ln(arcsin(x)) +ln(cos(x) + 2)▯ x ▯ ln x + x 2 0 ▯ ▯ 2 y = 1 p 1 ▯ sin(x) ▯ 2x ▯ 3x + 1 y arcsin(x) 1 ▯ x2 2(cos(x) + 2) x + x ▯ ▯ ▯ 2 ▯ 0 1 1 sin(x) 3x + 1 y = p 2 ▯ ▯ 2x ▯ 3 y arcsin(x) 1 ▯ x 2(cos(x) + 2) x + x ! ▯ ▯ ▯ 2 ▯ p = 1 p 1 ▯ sin(x) ▯ 2x ▯ 3x + 1 arcsin(x) cos(x) + 2 arcsin(x) 1 ▯ x2 2(cos(x) + 2) x + x e (x + x) 3. For each of the following equations, ▯nd the indicated derivative. 4y (a) 4x + y ▯ = 10x , ▯nd y . x + 3 We perform implicit di▯erentiation 0 2 0 4y (x + 3) ▯ 4y
More Less

Related notes for Mathematics 0110A/B

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit