Oh damn... I need paper
And calculator...
Okay, (a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.[I'm not writing the functions, it's early and I'm lazy, ha.]
fnInt(R(t),x,0,6)≈ 31.816 cubic yards
(b) Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.
Y(t)=S(t)-R(t)+2500
(c) Find the rate at which the total amount of sand on the beach is changing at time t=4.
What is this? rate of change... instantaneous? I think I forgot how to do this.
Wait a minute. Change of y over change of t at 4
So... Y(4)/4≈2498.091/4=624.52275 cubic yards per hour
(d) For 0≤t≤6, at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
Find critical points S'(t)-R'(t)=0 at t=3.007 and t=.505
t=3.007, S(t)-R(t)=-2.49
t=.505, S(t)-R(t)=-.253
endpoint values
t=0, S(t)-R(t)=-2
t=6, S(t)-R(t)=2.11
So absolute minimum at t=3.007
I think I got most of them wrong... If they are, someone explain please?
Well now, off to do the rest of the calculus homework... woohoo... homework...
Sunday, April 4, 2010
Saturday, March 6, 2010
Mean Value Theorem
f'(c) = [f(b)-f(a)]/(b-a)
1. This means that if you take a function, f(x)
then the slope of the secant line between two points, [f(b)-f(a)]/(b-a)
is equal to the slope of the tangent line at the point c, f'(c)
So for example:
f(x)=x²+1 (to make things easier to see)
f'(x)=2x
[f(b)-f(a)]/(b-a) where a=-2 & b=2 ==> [4-4]/(2+2)=0
f'(c) = [f(b)-f(a)]/(b-a) ==> 2c=0 ==> c=0
Basically, it means the the secant line and the tangent line are parallel to each other.
2. This only works for continuous and differentiable functions because if the function is neither, then either the point c or the slope at c may not exist, as in the function
f(x)=|x| where a=-2 and b=2 (sorry, no graph this time). In this case, the point c would have to be at 0, but since the function is not differentiable at 0, f'(c) does not exist and the mean value theorem does not apply.
1. This means that if you take a function, f(x)
then the slope of the secant line between two points, [f(b)-f(a)]/(b-a)
is equal to the slope of the tangent line at the point c, f'(c)
So for example:
f(x)=x²+1 (to make things easier to see)
f'(x)=2x
[f(b)-f(a)]/(b-a) where a=-2 & b=2 ==> [4-4]/(2+2)=0
f'(c) = [f(b)-f(a)]/(b-a) ==> 2c=0 ==> c=0
Basically, it means the the secant line and the tangent line are parallel to each other.
2. This only works for continuous and differentiable functions because if the function is neither, then either the point c or the slope at c may not exist, as in the function
f(x)=|x| where a=-2 and b=2 (sorry, no graph this time). In this case, the point c would have to be at 0, but since the function is not differentiable at 0, f'(c) does not exist and the mean value theorem does not apply.
Wednesday, February 17, 2010
f(x) from f'(x)
Woohoo... MORE graphs... (Yes, people, that was sarcasm). Okay, ANYWAYS...
1. f(x) increasing on interval [-2, 2] because f'(x)>0
f(x) decreasing on interval (-∞, -2]U[2, ∞) because f'(x)<0
2. Extrema are at x=-2--> minimum, because f'(x) changes from negative to positive and at x=2--> maximum, because f'(x) changes from positive to negative.
3. For the sake of simplicity I'll round to the nearest whole number...
Concave up on interval (-∞, -1]U[0, 1] because f"(x), aka the slope of f'(x), is positive
Concave down on interval [-1, 0]U[1, ∞) becasue f"(x) is negative
4. I want to say that f(x) might be somthing like -x^5
since it looks somewhat like -x^3 with a bunch of crazy stuff happening in the middle...
1. f(x) increasing on interval [-2, 2] because f'(x)>0
f(x) decreasing on interval (-∞, -2]U[2, ∞) because f'(x)<0
2. Extrema are at x=-2--> minimum, because f'(x) changes from negative to positive and at x=2--> maximum, because f'(x) changes from positive to negative.
3. For the sake of simplicity I'll round to the nearest whole number...
Concave up on interval (-∞, -1]U[0, 1] because f"(x), aka the slope of f'(x), is positive
Concave down on interval [-1, 0]U[1, ∞) becasue f"(x) is negative
4. I want to say that f(x) might be somthing like -x^5
since it looks somewhat like -x^3 with a bunch of crazy stuff happening in the middle...
Friday, January 15, 2010
Mindsets
Oh wow... that was interesting...
1. First off, this seems like a load of bull to me. I don't know about the rest of you, but my "mindset" changes depending on the subject/topic,it depends on my interest in it, not on whether or I believe the brain is a muscle that can expand or not. Hell, my brain is the last thing I consider when I'm faced with something new. The first thing I think is, "Will this be interesting/useful?" If not, I tend to go towards the fixed mindset, avoid challenges, obstacles and effort. If I do think the topic is interesting or useful, I go towards the growth mindset, I take the challenges as they come and put in most of my effort --- I say most because it's hard for a lazy person to ever put in ALL their effort. In other words, I don't think there is one specific mindset that a person, or at the very least, I, have. It depends on the activity, on whether it interests you or not.
2. As said, I don't believe there is one specific mindset that controls how you learn or think all the time. Isn't this part of the reason we do well with some topics and not so well with others? Personally, I go back forth at times when learning lots of things because some are more helpful or interesting than others. The only reason I even pay attention in class is that I'm sure it will help me when it comes time to take the damned AP test. Once I deem something useless it leaves my mind for a while, until I have to review the topic again.
3. To the mind is muscle thing: My reaction was, "I already knew this." You don't take two years of biology and a year's worth of psychology without learning this. And a couple biological or psychological documentaries tend to include this fact, too...
4. It shouldn't affect me much. Considering I had heard the mind being a muscle thing before, and now, again. Oh, how I hate repetition... As for the mindset stuff, that really shouldn't affect me either considering I don't think it's entirely legitimate.
1. First off, this seems like a load of bull to me. I don't know about the rest of you, but my "mindset" changes depending on the subject/topic,it depends on my interest in it, not on whether or I believe the brain is a muscle that can expand or not. Hell, my brain is the last thing I consider when I'm faced with something new. The first thing I think is, "Will this be interesting/useful?" If not, I tend to go towards the fixed mindset, avoid challenges, obstacles and effort. If I do think the topic is interesting or useful, I go towards the growth mindset, I take the challenges as they come and put in most of my effort --- I say most because it's hard for a lazy person to ever put in ALL their effort. In other words, I don't think there is one specific mindset that a person, or at the very least, I, have. It depends on the activity, on whether it interests you or not.
2. As said, I don't believe there is one specific mindset that controls how you learn or think all the time. Isn't this part of the reason we do well with some topics and not so well with others? Personally, I go back forth at times when learning lots of things because some are more helpful or interesting than others. The only reason I even pay attention in class is that I'm sure it will help me when it comes time to take the damned AP test. Once I deem something useless it leaves my mind for a while, until I have to review the topic again.
3. To the mind is muscle thing: My reaction was, "I already knew this." You don't take two years of biology and a year's worth of psychology without learning this. And a couple biological or psychological documentaries tend to include this fact, too...
4. It shouldn't affect me much. Considering I had heard the mind being a muscle thing before, and now, again. Oh, how I hate repetition... As for the mindset stuff, that really shouldn't affect me either considering I don't think it's entirely legitimate.
Saturday, December 19, 2009
Algebra vs Calculus
1. What is the DIFFERENCE between finding the limit of a function at x = c and actually plugging in the number x = c? When are the two cases the SAME?
Difference- when you are finding the limit you are finding the y value that wouldbe at x=c, if there was no discontinuity in the function. But, when you actually plug in the number x=c, you will get the output.
For example if there is a removable discontinuity at x=2 in a function, the limit might turn out to be 1, while if you plug in 2 into the function you might get an output of 3.
Same- the two are the same when there is no discontinuity at x=c and the limit is the same as the output of the function.
2. What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?
Similarities- the derivative of a line and the slope of the line are basically the same thing: the change in y over the change in x.
Differences- I'm fairly sure that the differences in the two are in where the derivative and the slope are used.
For example, finding the slope of a parabola, would give you the value of the slope of the parabola's secant line, while finding ther derivative of that same parabola would give you the slope of the parabola's tangent line.
Difference- when you are finding the limit you are finding the y value that wouldbe at x=c, if there was no discontinuity in the function. But, when you actually plug in the number x=c, you will get the output.
For example if there is a removable discontinuity at x=2 in a function, the limit might turn out to be 1, while if you plug in 2 into the function you might get an output of 3.
Same- the two are the same when there is no discontinuity at x=c and the limit is the same as the output of the function.
2. What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?
Similarities- the derivative of a line and the slope of the line are basically the same thing: the change in y over the change in x.
Differences- I'm fairly sure that the differences in the two are in where the derivative and the slope are used.
For example, finding the slope of a parabola, would give you the value of the slope of the parabola's secant line, while finding ther derivative of that same parabola would give you the slope of the parabola's tangent line.
Wednesday, December 9, 2009
Limits...
Okay well... limits...
Let's see... I think I understood most of the concept of limits. After going through the homework assignments of the past two weeks, I couldn't find any problems that I had left blank... Well, there was one, but after looking at it, I realized I could've just plugged in the number to find the limit. (It was #8 on pg 66, I think the large exponent threw me off haha)
I don't think there was much else...
Wait, one thing: Is there a difference between a limit being undefined and a limit not existing? I thought you could use the terms interchangabley but now I'm not so sure... Anyways, if there is a difference, what is it?
On a seperate note: How can you tell what a graph looks when the equation is in the form of a polynomial divided by a polynomial?
Ex: How can I know what
4x³-2x²+5
3x-4
looks like without having to use a calculator?
Let's see... I think I understood most of the concept of limits. After going through the homework assignments of the past two weeks, I couldn't find any problems that I had left blank... Well, there was one, but after looking at it, I realized I could've just plugged in the number to find the limit. (It was #8 on pg 66, I think the large exponent threw me off haha)
I don't think there was much else...
Wait, one thing: Is there a difference between a limit being undefined and a limit not existing? I thought you could use the terms interchangabley but now I'm not so sure... Anyways, if there is a difference, what is it?
On a seperate note: How can you tell what a graph looks when the equation is in the form of a polynomial divided by a polynomial?
Ex: How can I know what
4x³-2x²+5
3x-4
looks like without having to use a calculator?
Tuesday, November 24, 2009
Colleges
Majors
Preveterinary studies- not offered in many schools so you usually have to take other classes like biology and whatnot in order to be accepted into a veterinary school. Obviously, help prepare for the path to becoming a veterinarian. Need people skills...
Zoology- study animals. All kinds, not just vertebrates so probably not a good choice for those that a bit squeamish around bugs and worms and such. Study the animal and its anatomy and physiology like the cells and the organs. (Good references I bet...) A bit like ecology in that you study the animals' environment and how they adapt to it.
Wildlife biology- Study wildlife. The animals, the plants, the environment and how they all interact. Study the ecosystems from urban, suburban, rural areas and in the wild. Focus on vertebrates and their environments.
Colleges
UC Davis- In northern California, 15 miles from Sacramento. It has an equestrian and raptor center (raptor as in birds of prey, not the dinosaurs)
American University- Private school in Washington D.C. 80% of the students admitted are from out of state and offers AH DAMN IT! It's religiously affiliated....
Kalamazoo College- In Michigan. Only 32% of students from out of state were admitted and all first year students live in on-campus housing. But don't you just love the name. Kalamazoo. Haha...
Northern Arizona University- rural setting, 74% of applicants are admitted, 30% are from out of state. First-time first-year students are allowed to have a car. Located 140 miles from Phoenix, Arizona and there's a 400 acre forest close by as well as an observatory.
Stupid Amercan University... it was doing so well too... Oh well. The others are just as well.
Preveterinary studies- not offered in many schools so you usually have to take other classes like biology and whatnot in order to be accepted into a veterinary school. Obviously, help prepare for the path to becoming a veterinarian. Need people skills...
Zoology- study animals. All kinds, not just vertebrates so probably not a good choice for those that a bit squeamish around bugs and worms and such. Study the animal and its anatomy and physiology like the cells and the organs. (Good references I bet...) A bit like ecology in that you study the animals' environment and how they adapt to it.
Wildlife biology- Study wildlife. The animals, the plants, the environment and how they all interact. Study the ecosystems from urban, suburban, rural areas and in the wild. Focus on vertebrates and their environments.
Colleges
UC Davis- In northern California, 15 miles from Sacramento. It has an equestrian and raptor center (raptor as in birds of prey, not the dinosaurs)
American University- Private school in Washington D.C. 80% of the students admitted are from out of state and offers AH DAMN IT! It's religiously affiliated....
Kalamazoo College- In Michigan. Only 32% of students from out of state were admitted and all first year students live in on-campus housing. But don't you just love the name. Kalamazoo. Haha...
Northern Arizona University- rural setting, 74% of applicants are admitted, 30% are from out of state. First-time first-year students are allowed to have a car. Located 140 miles from Phoenix, Arizona and there's a 400 acre forest close by as well as an observatory.
Stupid Amercan University... it was doing so well too... Oh well. The others are just as well.
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