Definitions
Definition of derivative f at a
f'(a)=\lim_(x->a)\frac(f(x)-f(a))(x-a)
\lim_(x->infty)\frac(5x^2sin(1/x)+7x^3)(8x^+5x^2+3x^3)
7/3
\frac(d)(dx)[\int_(3x^2)^sinx(\sqrt(x^2+cos(x)))dx]
[\sqrt(sin^2x+cos(sinx))](cosx)-[\sqrt((3x^2)^2+cos(3x^2))](6x)
\int(sec^2(3x)+5x^6+3sinx)dx
1/3tan(3x)+5/7x^7-3cosx+C
d/dx[\int_g(x)^(h(x))f(t)dt]=f(h(x))h'(x)-f(g(x))g'(x)
Fundamental Theorem of Calculus - chain rule edition
Definition of f being continuous at a
\lim_(x->a)f(x)=f(a)
\lim_(x->5^-)\frac(|x-5||x-2|)(x^2-8x+15)
-\frac(3)(2)
\frac(d)(dx)(x^(sin(x)))
x^(xsinx)(cosxlnx+\frac(sinx)(x))
\int\d/dx((cos^5(x))/(4x^x))
(cos^5(x))/(4x^x)+C
Let f(x) be continuous on (a,b) . Then for any y between f(a) and f(b), there exists a c between a and b such that f(c)=y
The Intermediate Value Theorem
What is this process of simplification?
x^2+6x=1=>x^2+6x+9=10
(x+3)^2=10
Completing the square
\lim_(x->1)\frac(e^(2x-2)-1)(3x-3)
2/3
\frac(d)(dx)[\int_-6^9(9x^7+2sec^2x+(lnx)/x)dx]
0
\int\frac(cos(\sqrt(x)))(2\sqrt(x))dx
sin(\sqrt(x))+C
Suppose that f(x)\leqg(x)\leqh(x) and \lim_(x->a)f(x)=c=\lim_(x->a)h(x) . Then \lim_(x->a)g(x)=c
The Squeeze Theorem
What is this commonly used notation called?
\frac(dx)(dt)
Leibniz notation
\lim_(x->0)sin(1/x)
DNE
d/dx[e^(e^(e^(e^x)))]
e^(e^(e^(e^x)))e^(e^(e^x))e^(e^x)e^x
\int_-27^27(4x^7+(3x^3)^3+sin(x))dx
0
A continuous function on a closed interval [a,b] must have a max and a min. Another name for the min/max theorem.
What do we use to approximate integrals?
\lim_(x->infty)\sqrt(x^2+x)-x
1/2
A paper water cup has radius 3 centimeters and height of 7 centimeters, and is in the shape of a cone. The water is 4 centimeters deep, and the cone is placed such that the base of the cone is below the tip of the cone. The water's temperature is increasing at 4 millikelvins per second, and the water is dyed green. What is the acceleration of the top of the water?
-9.8 m/s^2
\int_-3^3\sqrt(9-x^2)dx
(9\pi)/2
If f is continuous on [a,b] , then there exists c\in[a,b] such that
f(c)(b-a)=\int_a^bf(x)dx
The Mean Value Theorem for integrals