WORKSHEETDISCUSSION SECTION 1DUE 4/5 AT MIDNIGHT

(1) Compute the following derivatives.

(a) Computeddxx

nusing the power rule.

(b) Computeddx(x

n�1 · x) using the product rule and the power rule for xn�1.

(c) Computeddx(x

n+1/x) using the quotient rule and the power rule for xn+1.

(d) Computeddxa

x, where ax = ex ln a

.

(e) Computeddxx

x, where xx

= ex lnx.

(2) Consider the function f : [0, 1] ! R, defined by f(x) = x+ x2+ x3

+ x4.

(a) Explain why f(a) = 2 for some 0 < a < 1.

(b) Give two reasons why f 0(b) = 4 for some 0 < b < 1.

Hint: use the intermediate value theorem and mean value theorem.

1

a nxn l

b n 1 x 2 x t x I I nxIntlc x ntl x x I nx n n 12 x2d a e ka Ina a In a

e Idi e h µIn x x Inx I

a f is continuous on 10 I and fO O f l L Since 2c O dtheintermediatevalue 1hm yields on a eLO L s t flat 2

b L f I 2x 3 2 43 so f O I f 1 10 Since f zcontinuous identical

reasoning to theaboveshows that there is

a bc O l s t f b 42 f is continuous on 10,11 and differentiable in 0 1 so bythemeanvaluethen thereis a be10 1 s t

Hb Ito to d

2 WORKSHEET DISCUSSION SECTION 1 DUE 4/5 AT MIDNIGHT

(3) A complex number is any number of the form z = x + iy, where x and y are real

numbers, and i =p�1. We write x+ iy = a+ ib if x = a and y = b.

(a) Find the complex solutions of x2+ 4 = 0

(b) Find the complex solutions of x2+ 2x+ 2 = 0.

Euler’s formula is: ei✓ = cos(✓) + i sin(✓). It follows from comparing power series.

(c) Show that ei⇡ = �1.

(d) Expand both sides of ei(x+y)= eix · eiy to derive the formulas

cos(x+ y) = cos(x) cos(y)� sin(x) sin(y) and sin(x+ y) = cos(x) sin(y) + sin(x) cos(y).

This is one way to re-derive trig identities on the fly. (I never remember them.)

a x 2 qbythequadrateformula

2 tb x2t 2x 2 0 I Ft2111

I i

c ei't cos it i sin at I Oi Ld e e f cosx isin x cosy ti schy

cos cosy sin xsing i sin x cosy cos x sisyeit P cos x t g t i sink gso matchingveal imaginarypartsgives

cos x yl cos x cosy sin x singsin x y sinx cosy cos x sinyas desired