SYMC Calculus AB

Theorems

100

If function is continuous on [a,b] then f must attain absolute maximum and minimum once

What is Extreme Value Theorem?

200

If a(x) <= f(x) <= g(x) for all numbers and at some point where x=k where a(k) = g(k), then f(k) must be equal at a(k) and g(k). 

What is squeeze theorem?

300

If f(x) continuous on [a,b], let M be between f(a) and f(b). Then there exists a point c where a < c < b and f(c) = M. 

What is Intermediate Value Theorem?

400

If f and F continuous on [a,b] and F'(x)=f(x), then the integral from a to b of f(x)dx = F(x)]^b_a = F(b)-F(a).

What is Fundamental Theorem of Calculus?

500

If f is continuous on [a,b], there exists a number c where the integral from a to b of f(x)dx = f(c)(b-a)

What is Mean Value Theorem for Integrals?

Derivatives

100

d/dx of xⁿ

What is nx^n-1? 

200

Derivative of sin(u)

What is cos(u)*u'?

300

d/dx ( ln(u) )

What is 1/u * du/dx?

400

d/dx ( cot (u) ) 

What is -csc²(u)*u'?

500

d/dx( sin^-1(x) )

What is ( 1/sqrt(1-x²) ) * x'?

Integral

100

Integral of 0dx

What is c?

200

Integral of e^x

What is e^x + c?

300

Anti-derivative of xⁿ?

What is (xⁿ⁺¹ / (n+1)  ) + c?

400

Anti-derivative of csc²(x)dx

What is -cot(x)+c?

500

Anti-derivative of  du / a²+u²

What is 1/a * tan^-1(u/a) + c?

Graph Analysis

100

f''(x) < 0

What is concave down?

200

f'(x) = 0 or does not exist

What is critical point?

300

point where a function changes continuity

What is point of inflection?

400

lim_x->a-

What is x approaches a from the left?

500

limit _x->a f(x) exists

f(a) exists

limit _x->a f(x) = f(a)

What is definition of continuity?

Anything

100

d/dx(e^x)

What is e^x * x'? 

200

lim_x->a f(x+h)-f(x) / h

What is the Definition of a Derivative?

300

integral from a to a of f(x) 

What is 0?

400

integral from a to b of |v(t)| represents

what is distance?

500

Formula for Washer Method

What is pi * integral of a to b of (R²-r²)