Calculus QuickSheet

MATH 119

1 Linear Approximations

L(x) = f(a) + f0(a)(x−a)

2 Root Finding

Bisection

Find a, b where f(a)<0 && f(b)>0.

•If f(a+b

2)<0 then a=a+b

2.

•Else, b=a+b

2.

Newton’s Method

1. Use Bisection to ﬁnd an estimated starting point (a).

2. Set L(x) = f(a) + f0(a)(x−a) to 0 and solve for x.

3. Let a=x.

4. Repeat steps 2-3 until you ﬁnd an acceptable estimate.

Iterative Formula: xn+1 =xn−f(xn)

f0(xn)

Fixed-Point Iteration

1. Isolate for x.

2. Solve xn+1 =g(xn).

3. Repeat until answer is same twice in a row (to required decimal places).

4. If divergence occurs, try isolating x diﬀerently and repeat steps 2-3.

1

3 Polynomial Stuﬀ

Newton Forward Diﬀerence Forumla

y=y0+(x−x0)

h∆y0+(x−x0)(x−x1)

2!h2∆2y0+... +(x−x0)(x−x1)...(x−xn+1)

n!hn∆ny0

Lagrange Linear Interpolation Formula

f(x)≈(x−x1)

(x0−x1)f1+(x−x0)

(x1−x0)f2

Taylor Polynomials

Pn,x0(x) =

n

X

k=0

f(k)(x0)

k!(x−x0)k+Rn(x)

Maclaurin Polynomial

Taylor Polynomial centered at 0. Pn,0(x) =

n

X

k=0

f(k)(0)

k!xk+Rn(x)

f(x) = f(0) + f0(0) ·x+f00(0) ·x2

2! +f000(0) ·x3

3! +...

Remainder Theorem

|Rn(x)|=|Zx

x0

(x−t)n

n!f(n+1)(t)dt|

≤ |f(n+1)(t)||x−x0|n+1

(n+ 1)!

2

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