Derivatives
d/dx (4x2)
8x
f'(x) = 7x6
f(x) = x7 + C
d/dx(xn)
nxn-1
If a function is smooth, there’s at least one point where the instantaneous rate equals the average rate.
Mean Value Theorem
Limit
d/dx (7x3-2x2)
21x2-4x
f'(x) = 5x3/2
Average Rate of Change
f(b)-f(a)/b-a
A continuous function on a closed interval must have a highest and lowest value.
Extreme Value Theorem
How much are exams worth?
45%
d/dx (5e2x+9x6-2x3)
f'(x) = 2ex + sec x tan x
f(x)= 2ex + sec x + C
Limh->0(f(x+h)-f(x))/h
f'(x) or df(x)/dx
A continuous graph hits every value between two points.
Intermediate Value Theorem
Where was Mr. Alfonso teaching before coming to SAS?
DASH
d/dx (4x3 sin x)
12x2 sin x + 4x3 cos x
f'(x)= (12/x5)
f(x) = -3/x4 + C
d/dx [loga(x)]
1/x ln(a)
Differentiation and integration are inverse operations.
Fundamental Theorem of Calculus
What was Mr. Alfonso's major in college?
Chemistry
d15/dx15 (cos x)
sin x
f'(x) = 9/(1-x2)1/2
9arcsin(x) + C or -9arccos(x) + C
Trapezoidal Approximation
delta(x)/2 * (f(xleft) + (xright))
If two functions trap another function and meet at the same limit, the trapped function has that limit too.
Squeeze Theorem
Who does Mr. Alfonso main in Mario Kart 8?
Toad