WORKSHEET DISCUSSION SECTION 1 DUE 4/5 AT MIDNIGHT (1) Compute the following derivatives. (a) Compute d dx x n using the power rule. (b) Compute d dx (x n-1 · x) using the product rule and the power rule for x n-1 . (c) Compute d dx (x n+1 /x) using the quotient rule and the power rule for x n+1 . (d) Compute d dx a x , where a x = e x ln a . (e) Compute d dx x x , where x x = e x ln x . (2) Consider the function f : [0, 1] ! R, defined by f (x)= x + x 2 + x 3 + x 4 . (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 nx Intl c x ntl x x I nx n n 1 2 x2 d a e ka Ina a In a e Idi e h µ In x x In x I a f is continuous on 10 I and f O O f l L Since 2 c Od the intermediatevalue 1hm yields on a e LO L s t flat 2 bL f I 2x 3 2 4 3 so f O I f 1 10 Since f z continuous identical reasoning to the aboveshows that there is a b c O l s t fb 4 2 f is continuous on 10,11 and differentiable in 0 1 so by the mean value then there is a be 10 1 s t Hb Ito to d
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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
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