Thank you very much Office_Shredder! One question though: I thought that I had to prove it in both orders, since it is an if and only if statement. Your route certainly seems more sensible but if I were to write the complete solution, wouldn't I assume in one case that as h goes to zero f(c+h)...
Homework Statement
Prove that limx\rightarrowcf(x)=L if and only if limh\rightarrow0f(x+h)=L.
Homework Equations
The Attempt at a Solution
I think this is a simple problem, but I am getting caught up in the middle, as I'm not sure if my procedure is a valid way to prove the...
Ok so I worked through all the exercises up to this one. Here's what I came up with.
prove:
A−(B\capC)=(A−B)\cup(A−C)
Let S=B\capC={xlx\inB, and x\inC} (By the definition of intersection)
Thus A-S={xlx\inA and x\notinS}
={xlx\inA,x\notinB, and x\notinC} (see justification...
Here's the proof for m=1
(an)m=(an)1=(By the definition)an
Now for m+1
Because xn+m=xn*xm
Let x=(an)
(x)(m+1)=(x)m*(x)1 and by the hypothesis and the case m=1 I have already proved and replacing x with (an)
(x)m*(x)1=(an)m*(an)1=anm*an=anm+n=an(m+1)
I'm pretty sure this proves...
an(m+1)=an*(m+1)=(by the theorem) (an)(m+1)
How else could I move from the left side of the equality to the right? If someone could give me a little direction I would love to solve it myself.
Homework Statement
Prove (a^{n})^{m}=a^{nm}
Homework Equations
Proof by induction
a^{n}*a=a^{n+1}
a^{n}*a^{m}=a^{nm}
The Attempt at a Solution
Let a and n be fixed. I will induct on m.
Suppose m=1. Then a^{(n)(m)}=a^{n(1)}=(a^{n})^{1}
Now assume the hypothesis is true for...
I don't see any alternate definitions of a/b. I only have these six axioms:
AXIOM 1. COMMUTATIVE LAWS. X +y =y + X, xy = yx.
AXIOM 2. ASSOCIATIVE LAWS. x + (y + z) = (x + y) + z, x(yz) = (xy)z. AXIOM 3. DISTRIBUTIVE LAW. x(y + z) = xy + xz.
AXIOM 4. EXISTENCE OF IDENTITY ELEMENTS. There...
Homework Statement
Prove: if a\neq0 then b/a=b*a^-1
I don't know if my proof is sufficient with this one but I cannot think of another way. Also I am only supposed to use the basic axioms (e.g. commutative, distributive, existence of reciprocal...)
Homework Equations
1/a=a^-1 (I am...
A-(B\bigcapC)=(A-B)\bigcup(A-C)
If A-B={xlx\inA and x\notinB}
A-C={xlx\inA and x\notinC}
then (A-B)\bigcup(A-C)={xlx\inA, x\notin(B and C)
Let X=A and Y=(B\bigcapC)
X-Y={xlx\inX and x\notinY}
x\notinY
x\notin(B\bigcapC)
x\notin(B and C)
Therefore...
oops I meant "Rate of change of area of a square with respect to its side length"
Ok I have to use this annoying Stewart textbook for my Calc class in college. Most of the questions require what I like to call "Monkey Math," where you just memorize a set of steps and then follow them rigidly...
did you get something like 30.852 km/h for your final answer? I tried this out and got that. Although one thing I didn't understand was what the angle was given for. Because how is that relevant to the speed of the car upon entering the crossing? Unless you know the length of the train engine...
Hello PF,
I am a Sophomore in high school in the US and have already led what I would consider to be a pretty awesome life, but I have an interesting dilemma.
All my life I have loved to ski, I ski competitively, and I believe that if I continue to pursue skiing I could possibly become a...
Hmmm, I worked through that and I got something still around 2000 pounds.
I think the problem with that equation may lie that you are subtracting the horizontal velocity, whereas shouldn't the horizontal velocity be combined with the angle of the slope to somehow find a percentage of the...