Calc Handbook

The tangent line’s slope = derivative

Normal line is perpendicular to the tangent line

Linear approximation - L(x) = f(a) + f'(a)(x - a).

Exponential growth and decay: P(t) = P(0)ek

ek is constant so it can just be substituted for d (when you find ek, it’s different for every problem)P(t) = P(0)d^t

Half life: h(t) = h (0 )2t

Newton’s law of cooling: T(t) = (T(0) – Ts)e^kt +TsWhere Ts is the temperature of the surroundings

E limit definition –

Newton’s method –

Partial fractions – split up the fraction so it is easier to integrate x - 1 A B --------------- = -------- + ------- (3x - 5)(x - 3) (3x - 5) (x - 3)

then cross multiply and set the numerators equal to the integral numerator

x - 1 = A(x - 3) + B(3x - 5)

when you multiply it all out, set the x terms equal to the x coefficient and the constants equal to the constants

if there is a squared term

1 A B C --------------- = ------- + ------- + -------- (x - 3)(x + 1)² (x - 3) (x + 1) (x + 1)²

if there is an unfactorable quadratic

x - 1 A Bx + C -------------------- = --------- + ------------- (x + 3)(x² + 3x + 2) x + 3 x² + 3x + 2

Arc length :

Hookes Law – force required to maintain a spring stretched x units beyond its natural length is proportional to x:

F(x) = Kx where k is the spring constant

The dot product – the product of two vectors (equals a scalar)If a and b are vectors, then A B = axbx + ayby + azbz

A B = |a||b| cos (to solve for theta)

The cross product – the resultant vector of 2 or more vectorsA = <1,2,3> b = <4,5,6>

Then:A x b = | I j k |

| 1 2 3 | | 4 5 6 |

= I |2 3| - j |1 3| + k |1 2| |5 6| |4 6| |4 5|

= i(2x6-3x5) – j(1x6-3x4) + k(1x5-2x4)= <-3, 6, -3>

multiplying the cross product by the dot product of one of either a or b will result in zero every time since the cross product is orthogonal to both vectors

Vector projections:

length of L = |A|cos composition of a onto b = vectorA∗vectorB[VectorB ](gives you a scalar number)

the projection of a onto b = vectorA∗vectorB[VectorB ] *vectorB|B|

where vectorB|B| is the direction of the vector(gives you a new vector with direction and magnitude)

a vector is parallel to another if it is the same vector multiplied by a constant

the projection of b onto a is also parallel to aa vector is orthogonal if the dot product is 0a vector N is orthogonal to a to a vector b minus the projection of b onto a

Work and Forcemultiply the mass density of water, 1000 kg/m^3, by the force due to gravity, 9.8 m/s^2

Vector AreaArea of a triangle = ½|vectorA x vectorB|

Area of a parallelogram = |vectorA x vectorB|

Area of a Parallelepiped = |vectorA (vectorB x vectorC)|

Vector Equation: vectorR = vectorR0 +t*vectorV

Parametric equation: x = x0 +at y = y0 +bt z = z0 +ct

Symetric Equations: (x-x0)/a (y-y0)/b (z-z0)/c

Logs

Disks

Function of xIf the function to be revolved is a function of x, the following integral represents the volume of the solid of revolution:

Function of y

If the function to be revolved is a

function of y, the following integral will obtain the volume of the solid of revolution:

Washers

where RO(x) is the function that is farthest from the axis of rotation and RI(x) is the function that is closest to the axis of rotation. NOTE: the above formula only works for revolutions about the x-axis.

To rotate about any horizontal axis, simply subtract from that axis each formula:

if h is the value of a horizontal axis, then the volume =

Shells

function p(x) is the distance from the axis and h(x) is the length of the shell, generally the function being rotated.

Approximations

Midpoint :∆x ¿

x i=12(x i−1+x i)

-Remember – Midpoint will have the same number of factors to add as the n value

Trapezoidal : ∆2

( f (x0 )+2 f (x1)+2 f (x2 )+ f (xn ))

-Remember – Trapezoidal will have 1 more factor to add than the n value

Simpsons : ∆3

(f (x0 )+4 f (x1 )+2 f (x2 )+...4 f (xn−1 )+ f (xn ))

-Remember – Simpsons will have 1 more factor to add than the n value