Calculus Fund

put in all the rules and all the defs, then critical vals yada yada

Limit Def

f’(x) lim h→0 = (f(x+h) - f(x))/h

Backward Difference

(f(x) - f(x-h))/h

Forward Difference

(f(x+h)-f(x))/h

Central Difference

(f(x+h)-f(x-h))/2h

Tangent Line Approx

f’(a)(x-a) + f(a)

Deriv of Sin and Cos

Sin → Cos, Cos → -Sin

Deriv of Arcsin

1 / √1-x²

If f(x) is concave down, what is our tangent line approx?

An over estimate

If f(x) is concave up, what is our tangent line approx?

An underestimate

What are zeros of a fxn?

When f(x) = 0

Where are critical numbers of f(x)

When f’(x) = 0 or is undefined

Product Rule d/dx [f x g]

f’ g + g’ f

Quotient Rule d/dx [f/g]

(g f’ - f g’)/g² low de-high - high de-low, all over low²

Deriv of Arctan

1 / 1+x²

Chain Rule d/dx [f(g)]

f’(g(x)) × g’

Natural Exponent e^x

e^x

Deriv of Natural Log ln(x)

1/x

Deriv of the Inverse, g is an inverse of f

g’ = 1/f’(g)

Exponential Rule

Power Rule

x^n → nx^n-1

Constant Rule

5x² →5(2x^1) → 10x …… along for the ride!

Related Rates Steps

1. Draw diagram and label variables that change respect to time (t)

2. List the rates given, and what rates you want with notation ( dr/dt | r=3)

3. Find the parent fxn, (sphere, cube, rectangle?)

4. Differentiate w/ respect to time and implicitly

5. Plug in knowns and solve for rate!

What is a related rates problem?

Given the rate of change at a certain moment, find other ROC’s at that same time

What happens when you differentiate a fxn of time?

r² → 2r dr/dt, it seperates into the value times the ROC

If f is CCU on an interval, what does this also say?

f’ is increasing on the interval, and f’’ is positive

If f is CCD on an interval, what does this also say?

f’ is decreasing and f’’ is negative <0

Given f’ or f’’, where are the inflection points of f?

If f’ is > 0

f is increasing

If f’ < 0

f is decreasing

If f’(x) = 0

x is a critical point

Quadratic Formula

(-b ± √b²-4ac ) / 2a