Computations
Differentiate the following
f(x)=e^{-2x^3}
f'(x)=-6x^2e^{-2x^3}
What does the meaning of the derivative? What is its geometrical interpretation?
The derivative tells us the instantaneous rate of change at a given moment and geometrically is equivalent to the slope of the tangent line at a point.
Find
A\cup B
A\capB
A={1,2,3}
B={2,c}
A\cup B={1,2,3,c}
A\cap B={2}
I say that 6 + 7 = 1. How can that be possible?
Modular arithmetic! 6 AM+ 7 hours= 1 PM think in clocks.
Evaluate the following limit
\lim_{x \to 0}\frac{\text{sin}6x}{x}
6
Does continuity imply differentiability? Why or why not?
Continuity does not imply differentiability. For example, a function may be continuous but may exhibit a cusp which would make it undifferentiable at that point.
Is the following proposition a tautology? Come up to the board and write down the truth table
(P\rightarrow Q)\wedge(Q \rightarrow P)
Correct truth table on board etc.
What is the sum of the first 100 natural numbers? For double the points, what was Gauss' method?
5050. Gauss made 50 pairs of 101 and quickly realized they give this result.
Evaluate the following limit
\lim_{x \to 2\pi}(-3x^2)(\text{cos}(\frac{1}{x^2})
DNE
Write down the chain of subsets with respect to the various number systems we learned about. i.e. reals rationals and so on...
\mathbb{N\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteqq \mathbb{R}\subseteqq \mathbb{C}}
In the following example is A a subset of B? Given the map defined what is its image, what is its inverse image? What kind of map is this
{x\in\mathbb{N:x=5\mathbb{N}}}
{x\in\mathbb{Z:x=5\mathbb{Z}}}
f:A\rightarrow B
f(5)=5
f(10)=10 ...
A is a subset of B. This is an injective map.
\text{Im}f={x\in\mathbb{N}:x=5\mathbb{N}}
f^{-1}(B)= {x\in \mathbb{N}:x=5\mathbb{N}}
Using only addition, add eight 8s to get the number 1,000
888+88+8+8+8=1000
Differentiate the following
e^x\text{ln}(x}+\frac{e^x}{x}+\frac{(2x)(2x^2-3)-(x^2+2)(4x)}{(2x^2-3)^2}
ANSWER ON BOARD DOES NOT FIT ON SLIDE
Come up to the board and draw the position vs. time, velocity vs. time and acceleration vs. time graph for a particle with position function *Write all corresponding derivatives.
f(t)=x^2
Prove the following proposition: P: If m is odd and n is odd, then there sum, m+n, is even
Correct proof.
Two fathers and two sons make wooden chairs. If each makes a wooden chair, why are there only three produced?
There are only three people – a father, his son, and his son’s son.
Find the derivative of the following using the limit definition of the derivative. *Show all work
f(x)=x^2-12x+35
f'(x)=2x-12
Come up to the board and sketch a graph which meets the following conditions
f(-1)=2
\lim_{x \to -1}f(x)=4
\lim_{x \to 1}f(x)>\lim_{ x\to 0}f(x)
f \text{is decreasing from} 1<x<3
Correct graph on board.
Prove the following proposition: P: If a divides b and c divides d, then ac divides bd
Correct proof.
Is it always true that if you double the sum of two squares you get the sum of two squares? If so can you prove it?
2(x^2+y^2)=(x+y)^2+(x-y)^2
=(x+y)(x+y)+(x-y)(x-y)
=x^2+2xy+y^2+x^2-2xy+y^2
=2x^2+2y^2
2(x^2+y^2)