Calculus is a math which is focused on limits, differential, and integrals and Isaac Newton is a person who invented calculus. People call calculus as a flower of math and it is because differential, and intefrals are very important in math. Today I am going to talk about derivatives.

The differential is the most basic thing in calculus. It is like the four fundamental arithmetic operations, so if you do not learn the differential, you cannot solve any questions in calculus. The differential is basically shows a slope of a line. We use the differential to find the slope because lines are not always straight, so we cannot use rise over run to find the slope of the line. What we can find by taking the derivative is a line called tangent line and it shows the rate of change in a specific part of the graph. Therefore, it is very useful mathematical skill.

The differential also can be applied to other subjects. It is able to find a velocity of an object by take a derivative of a speed of the object and we can get an acceleration of the object if we take the derivative again. Therefore, it is very important to students even if they do not take math.

A function to be analytic at c means that the function is differetiable for many times and the function has a convergent power series at the point c. For example, the absolute value function is not analytic because we cannot differentiate the function at x=0.

The analytical function is useful because it has a convergent power series at some point. Therefore, it is useful when we are solving the questions. Also, we can distinguish the analytical function by taking a derivative of the function at some point.

The most difficult part in the midterm was all the proving parts in few questions. The questions which asked me to explain or define were very hard to me. This was because I did not practice any problems about proving and defining. I had solved many questions about taking integration, integration by substitution, integration by parts, and finding volumes. However, it was little bit hard to practice prving things. Also, I spent too much time to write the poem to earn only one mark more. I had to spend more time on solving questions. I made few mistakes because I could not manage time wisely, so I was not cool enough to solve hard questions.

To make a good substitution, it is important to get rid of original letters like x or t in the function because we cannot integrate the function if there are two letters in the function.

Also, if there is a square root in the function, it is better to let u equals to a letter in the square root because it would be easier to integrate.

Finally, it is important to change the form by using trig identity. We learned that sin^2+cos^2=1 and tan^2+1=sec^2. Sometimes, it might be really helpful to change the form if sin^2, cos^2, tan^2, and sec^2 are in the function.

To check this function is integrable or not, what we have to do is use the definition of integral. If we can prove this function is integrable, we can explain why the functions with finitely many removable discontinuities are integrable. The first thinf we have to do is partition the interval into n subintervals and select any sample point in the subintervals. If the function is integrable, we can find the area under the curve and it should exist and be equal in every sample points. In the question, it mentioned that there are “finitely” many removable discontinuities. It means the number of the discontinuities are not many as the number of irrational numbers. If there were infinitely many number of discontinuities are in the interval, the function might not be integrable. However, in this question, there are only finite number of the discontinuities. Therefore, the discontinuities do not really affect to the area under the curve.

Question: Find the domain, intercepts, and asymptotes of f(x) when f(x)=(6(x)^2)+21x+15

Answer: Domain: x can be all real numbers

Intercepts: y-intercept exists when x=0, so it is (6(0)^2)+21(0)+15=15

x-intercept exists when y=0, so 0=(6(x)^2)+21x+15=3(2(x)^2)+7x+5)=3(2x+5)(x+1) Therefore, x-intercepts would be -5/2 and -1

Asymptotes: There are no vertical asymptotes

Also, no horizontal asymptotes.

A cleaning man is cleaning Koerner library with long cleaning equipment. The cleaner puts the equipment on the ground and the equipment 5 meters long leans against the library. If the bottom of the equipment slides away from the library horizontally at a rate of 0.5 m/s, how fast is the ladder sliding down the house when the top of the ladder is 3 meters from the ground?

What we know in this question is y=3, z=5, dx/dt=0.5 and the equation we have to use is       (x^2)+(y^2)=z^2

if we differentiate the equation, 2x(dx/dt)+2y(dy/dt)=2z(dz/dt) and we plug the numbers we already know.

2x(0.5)+2(3)(dy/dt)=2(5)(0), dz/dt equals 0 because it does not change.

To find x, we use (x^2)+(y^2)=z^2 and plug y=3, and z=5

Then x^2=25-9=16. Therefore x=4 and we plug 4 to 2x(0.5)+2(3)(dy/dt)=2(5)(0).

2(4)(0.5)+2(3)(dy/dt)=2(5)(0)

4+(6)(dy/dt)=0

(6)(dy/dt)=-4

dy/dt=-2/3

a) Let f(x) be continuous on  between numbers a and b. Then for any y between f(a) and f(b), there exists c between the numbers a and b. In this question, it claims that Pa is equal to the temperature at Pa+π by the Intermediate Value Theorem. The Intermediate Value Theorem works in the question because f(0) is smaller than 0 and f(π) is bigger than 0, or vice versa. Therefore, if there is a number a in between 0 and π, it means that there is f(a) between f(0) and f(π). When the function is continuous and there is a number between two other bigger and smaller numbers, it is impossible to the y-value of the number is bigger than y-value of the bigger number, and smaller than y-value of the smaller number in circular shaped graph. Therefore, there exists a number a in between 0 and π such that f(a) equals 0 and the temperature at Pa is equal to the temperature at Pa+π

b) My argument relies on the concept of continuity.

a) The length of shadow is a function of the object’s height and the angle of the sun. The longest height of the object is when the sun’s angle to the Earth is very low or very high. Therefore, the height gets lower as the time passes and it gets higher again after the sun gets lower after 12’o clock. However, the shadow does not get longer after some point because the sun sets and there is no shadow until the sun rises again. This function has horizontal asymptote because the height of the shadow can approach 0, but it cannot be lower than 0.

b) Radioactivity level is one of examples of sequences because the radioactivity level decreases constantly. If we say that the radioactivity level decreases 5% every year, the amount of decreasing would be getting smaller and the level cannot reach 0. However, the level would be really close to 0 after many years. Therefore, this sequence would converge to 0.

C)  Bank account is one of examples of series, but not all the bank accounts are examples of the series. If a person puts money in the account, the account would pay probably 5% of interest to the person. As time passes, the account would pay more interest even if the percentage of the interest does not change because if the person does not withdraw money in the account, the account must have more money than the beginning. If there is more money, it means there would be more interest. The series will converge to infinity because as the money in the account gets bigger, the interest would get bigger, too. Therefore, the series would be converge to infinity.

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