Calculus SI -2/15

Limit Strategies

100

Substitution

Plug that sucker in

200

Algebraic Cancellation

Crossing out terms that equal out to zero

300

Rationalization

Multiplying by the opposite with a square root 

400

The trig limits

lim_(x->0)(sin/x)=1

lim_(x->0)(1-cos/x)=0

500

The squeeze theorem

Assume that for all x does not equal c (in some neighborhood of c) If B(x)<f(x)<T(x) and              lim_x->cB(x)=L=lim_x->cT(x) then lim_x->cf(x)=L.

Formulas

100

Phnemonic for quotient rule

Hi-de-lo lo-de-hi all over lo-lo

200

The power rule (with x and n)

What is nx^(n-1)?

300

The product rule

f'(x)g(x)+f(x)g'(x)

400

The quotient rule

(g(x)f'(x)-f(x)g'(x))/(g(x)²)

500

The definition of a derivative

What is lim_(h->0)(f(a+h)-f(a))/(h)?

Power Rule

100

3

0

200

4x

4

300

7x³

21x²

400

4x²-5x

8x-5

500

6x⁶+e^x-3x²+7x+9

36x⁵+e^x-6x+7

Product Rule

100

(x²-3)(x-5)

(2x)(x-5)+(x²-3)(1)

200

x²e^x

2xe^x+x²e^x

300

(2t+1)(t²-2)

(2)(t²-2)+(2t+1)(2t)

OR

6t²+2t-4

400

(t³-7t)(4e^t)

(3t²-7)(4e^t)+(t³-7t)(4e^t)

500

sqrt(w)e^w

(-1/2)w^-1/2e^w+w^1/2e^w

Quotient Rule

100

(2x-6)/(x²-7)

((x²-7)(2)-(2x-6)(2x))/(x²-7)²

200

e^x/(x+1)

((x+1)(e^x)-(e^x))/(x+1)²

300

(2x³)/(3x²-7)

((3x²-7)(6x²)-(2x³)(6x-0))/(3x²-7)²

400

(6x²-1)/(x²+2)

((x²+2)(12x)-(6x²-1)(2x))/(x²+2)²

500

(3z+e^z)/(6z¹⁰-2)

((6z¹⁰-2)(3+e^z)-(3z+e^z)(60z⁹))/(6z¹⁰-2)²