math 52 midterm

hi

d/dx log_ax

1 / (ln a × x)

d/dx sin^-1(x)

1 / √ 1-x²

d/dx cos^-1(x)

-1 / √1-x²

d/dx sec^-1(x)

1 / |x| √x²-1

d/dx csc^-1(x)

-1 / |x| √x²-1

d/dx tan^-1(x)

1 / 1+x²

d/dx cot^-1(x)

-1 / 1+x²

integral of a^x

a^x / ln a +c

integral of tan x

-ln |cos x| +c

integral of cot x

ln |sin x| +c

integral of sec x

ln |sec x + tan x| + c

integral of csc x

-ln |csc x + cot x| + c

integral of log_ax

x ln x - x / ln a +c

left riemann sum formula

(width of each subinterval)(leftheight1 + leftheight2 + leftheight3…)

right riemann sum formula

(width of each subinterval)(rightheight1 + rightheight2 + rightheight3…)

midpoint riemann sum formula

(width of each subinterval)(middleheight1 + middleheight2 + middleheight3…)

trapezoidal riemann sum formula

(width of each subinterval / 2) (height1 + 2*height2 + 2*height3 + 2*height4 +… heightfinal)

simpsons rule formula

(width of each subinterval / 3) (height1 + 4*height2 + 2*height3 + 4*height4 + 2*height5 +… heightfinal)

∫_a^bf(x)dx =

-∫_b^af(x)dx

direct comparison test

when 0<an<bn, if bn converges, an converges. if an diverges, bn diverges (an is the smaller series, bn is the bigger series)

sin²(x) (power reduction)

1 - cos2x / 2

cos²(x) (power reduction)

1 + cos2x / 2

sin(2x) (double angle identity)

2sinxcosx

cos(2x) (double angle identity)

cos²x-sin²x
2cos²x - 1
1-sin²x

x = asinθ

sqrt(a² - x²)

x = atanθ

sqrt(a²+x²)

x = asecθ

sqrt(x²-a²)

sin²x + cos²x

1

tan²x + 1

sec²x

1 + cot²x

csc²x

error bound for midpoint sum

error ≤ K(b-a)³ / 24n² (K is based on 2nd derivative)

error bound for simpsons rule

error ≤ K(b-a)⁵ / 180n⁴ (K > is based on 4th derivative)

error bound for trapezoidal sum

error ≤ K(b-a)³ / 12n² (K is based on 2nd derivative)

1 / sin x =

csc x

1 / csc x =

sin x

1/ cos x =

sec x

1/ sec x =

cos x

sin x / cos x =

tan x

cos x / sin x =

cot x