Differentiating and Integrating Mathematical Functions: Solutions and Examples, Assignments of Calculus

Download Differentiating and Integrating Mathematical Functions: Solutions and Examples and more Assignments Calculus in PDF only on Docsity! Jim LambersMath 2BFall Quarter 2004-05Homework Assignment 5 Solution1. Di erentiate the following fun tions with respe t to x.(a) f(x) = 2sinxSolution Using the rule ddx [ax℄ = ax lna;in onjun tion with the Chain Rule, we obtainf 0(x) = 2sinx ln 2 ddx [sinx℄ = 2sinx ln 2 os x:(b) f(x) = log3(x22x). Di erentiate without using the Produ t Rule.Solution First, we use the properties of logarithmi fun tions,logb(xy) = logb x+ logb y; logb xy = y logb xto obtain f(x) = log3 x2 + log3 2x = 2 log3 x+ x log3 2:We then use the rule ddx [logb x℄ = ddx  lnxln b  = 1x ln bto obtain f 0(x) = 2x ln 3 + log3 2:( ) f(x) = sin1( s x)Solution Using the rulesddx [sin1 x℄ = 1p1 x2 ; ddx [ s x℄ = s x ot x;in onjun tion with the Chain Rule, we obtainf 0(x) = 1p1 s 2 x ddx [ s x℄ = s x ot xp1 s 2 x:1 (d) f(x) = se (tan1 x)Solution Using the rulesddx [se x℄ = se x tan x; ddx [tan1 x℄ = 11 + x2 ;in onjun tion with the Chain Rule, we obtainf 0(x) = se (tan1 x) tan(tan1 x) ddx [tan1 x℄ = se (tan1 x) x1 + x2 :The x in the numerator arises from the an ellation equation tan(tan1 x) = x.2. Evaluate the following de nite integrals. Please simplify your answer as mu h as possible.(a) R 20 x10x2 dxSolution Using the substitution u = x2, for whi h du = 2x dx, we obtainZ 20 x10x2 dx = 12 Z 40 10u du= 12 10uln 10 40= 12 ln 10(104 100)= 99992 ln 10 :(b) R 30 10log100 x2 dx. Hint: use the hange-of-base formula and the fa t that ax is the inverseof loga x. Then the problem will be ome very easy. Really.Solution First, we rewrite the integrand using the hange-of-base formulalogb u = loga uloga b ;with u = x2, b = 100, and a = 10. This yieldslog100 x2 = log10 x2log10 100 = 2 log10 x2 = log10 x:It follows from the an ellation equationblogb x = x2