calculus_13_Useful_Formulas

B

Useful Formulas

Algebra

Remember that the common algebraic operations have precedences relative to each other: for example,mulitplication and division take precedence over addition and subtraction,but are“tied”with each other.In the case of ties,work left to right.This means,for example, that1/2x means(1/2)x:do the division,then the multiplication in left to right order.It sometimes is a good idea to use more parentheses than strictly necessary,for clarity,but it is also a bad idea to use too many parentheses.

Completing the square:x2+bx+c=(x+b

4

+c.

Quadratic formula:the roots of ax2+bx+c are−b±√

2a

.

Exponent rules:

a b·a c=a b+c

a b

a

Geometry

Circle:circumference=2πr,area=πr2.

Sphere:vol=4πr3/3,surface area=4πr2.

291

292Appendix B Useful Formulas

Cylinder:vol =πr 2h ,lateral area =2πrh ,total surface area =2πrh +2πr 2.

Cone:vol =πr 2h/3,lateral area =πr

√r 2+h 2+πr 2.

Analytic geometry

Point-slope formula for straight line through the point (x 0,y 0)with slope m :y =y 0+m (x −x 0).

Circle with radius r centered at (h,k ):(x −h )2+(y −k )2=r 2.

Ellipse with axes on the x -axis and y -axis:

x 2b 2=1.

Trigonometry

sin(θ)=opposite /hypotenuse

cos(θ)=adjacent /hypotenuse

tan(θ)=opposite /adjacent

sec(θ)=1/cos(θ)

csc(θ)=1/sin(θ)

cot(θ)=1/tan(θ)

tan(θ)=sin(θ)/cos(θ)

cot(θ)=cos(θ)/sin(θ)

sin(θ)=cos π

2−θ

sin(θ+π)=−sin(θ)

cos(θ+π)=−cos(θ)

Law of cosines:a 2=b 2+c 2−2bc cos A

Law of sines:a

sin B =c

Appendix B Useful Formulas293 Cosine of double angle:cos(2x)=cos2x−sin2x=2cos2x−1=1−2sin2x Cosine of difference of angles:cos(x−y)=cos x cos y+sin x sin y

tan x+tan y

Tangent of sum of angles:tan(x+y)=

2

1+cos(2θ)

cos2(θ)=