I will explain what is Fundamental Theorem of Calculus in this reflection. Let a function f(x) be continuous on an interval from a to b, and F(x) be the indefinite integral of f(x). If F(x) is differentiable, then the derivative of F(x) will equal to f(x). Let G(x) be the antiderivative of f(x), then the integral of f(t) from r to l will equal to G(r) minus G(l).

With this knowledge, we can calculate area between functions and axises or between functions and functions. For example, if we want to calculate the area above the x-axis and under the function y = 2x − x^2, we should calculate the antiderivative of y first which is x^2 − 1/3(x^3). Then we need to calculate points when function y and x-axis intersect with each other which are  0 and 2. By Fundamental Theorem of Calculus, we plug x=2 and x=0 into y’s antiderivative x^2 − 1/3(x^3) to get the final answer 4/3.

Infinitely differentiable functions are also called as smooth functions in mathematics. Any analytic function of a real argument is smooth, but the converse is not true. An analytic function is a function given by a convergent power series. A function is analytic only if its Taylor series about x0 converges to the function for every x0 in its domain. Such as all polynomials, exponential function, trigonometric functions, logarithm, and the power functions.

The most difficult question for me on midterm is the fourth question. Although there is a hint, I did not get it. Because f”(t), f'(t) and f(t) made the question looks complicated, I was thinking to use FTC all the time but did not open my mind through the hint.

Study Tips:

1. Mark problems you feel uncomfortable when doing weekly webworks. Redo those questions before midterm.
2. Memorize definitions of theorems and redo classic examples from workshops or recitations.

3 tips for integraling:

Key 1:  ∫ sin(ax+b) dx = − 1/a cos(ax+b)+c     Such as  ∫ cos(3x+4) dx = 1/3 sin(3x+4) + c

Key 2:  ∫ 1/(ax+b) dx = 1/a ln|ax+b|+ c       Such as   ∫ 1/(3x+7)= 1/3 ln|3x+7|+ c

Key 3: When choose u and dv, For u, we should choose a simple one as u. we’d better think of du in order to integral dv easier.

Key 4: practice makes perfect

From the example provide from the question, we know when t is not equal to 1, 2, or 3, f(t) will be t^2. In this situation, we know when t gets larger, t^2 will get larger as well. This also implicates that the area will increase with t’s increasing. f(t)=t^2 is a kind of quick increasing function, the area will also increase fast. When the area reaches a huge enough number, the area of t=1, t=2, t=3 can be ignored. Because at that time, comparing the area of f(t1), f(t2), f(t3) and other areas, the area of f(t1), f(t2), f(t3) are pretty small, these three areas will not affect whether the function is integral or not. Even in a slow growth function, as t approaches to infinity, area increased definitely. Therefore, finite points’ areas are not important any more. This is why functions with finitely many removable discontinuities are integrable.

Question: A 5-meter length ladder leans against a wall. At the beginning, the ladder is at rest. Then bottom of the ladder is sliding away from wall by a rate of 0.2 m/s. What is the rate of the top of ladder when the bottom is 3 meters away from the wall?

Solution: Assume the length of ladder is L. The distance from the bottom to the wall is D. The distance from the top to the ground is H. Because the angle of wall is a right triangle, we can use Pythagoras Theorem L^2=D^2+H^2. From the question, we know L=5, D=3, therefore, H=sqrt(25-9)=4. Then take the derivative of equation, we get 2LL’=2DD’+2HH’. We know the length of ladder does not change, so L’=0. Also from the question, we know D’=0.2. Now we can plug known values into equation, we get 2(5)(0)=2(3)(0.2)+2(4)H’. Do the calculation: -4H’=0.6 therefore H’=-0.15m/s which means the top of ladder is decreasing 0.15 meters per second.

Last week, people blew lots of balloons for decorating the celebration arena. Suppose air was being pumped into each balloon at a rate of 4 cm³/sec. What is the rate at which the radius of the balloon is increasing when the radius of the balloon is 6 cm.

We know the volume of a sphere is V=4/3πr³. Then we do the derivative which is dv/dt=(4πr²)(dr/dt). From the question, we get r=8, dv/dt=3. Plug these numbers into the equation, we get 4=(4π6²)(dr/dt). Therefore, dr/dt=0.00884 cm/sec.

We can imagine a point P on a circle, and Pθ is a point obtained from P by rotating it around the circle by angle θ counterclockwise. Assume T(θ) is the temperature at Pθ, and T(θ) is continuous from 0 to 2π (which also included 0 and 2π), and T(0) equals T(2π). Then assume another function f(θ)= T(θ) − T(θ + π). If values of θ are from 0 to π, so f(θ) is the temperature’s differences of two opposite points P and Pθ. Now we have two possibilities, if f(0) equals 0, then f(0) equals f(θ) equals 0. But if f(0) is not equal to 0, then f(p)=T(π)-T(2π). As we mentioned before, T(0) equals to T(2π). So, f(π)=T(π)-T(0). We can also write as f(π)=-(T(0)-T(π)) which is –f(0). Therefore, if f(π) is positive, then f(0) is negative (vise versa). By the Intermediate Value Theorem, if there is a point c larger than zero but smaller than π, then f(0) equals to zero. Therefore, T(c)=T(c+π). C and C+π have the same temperature but opposite positions. We can also imagine these two points are two countries on the earth. We know the countries locate on the equator of earth has the highest temperature. Earth is a kind of sphere, countries are on the equator have the same temperature but can be the opposite to each other.

1. For a function: When I toss a coin up, the trend of the coin will be a parabola function. To assume the x-axis is time, and y-axis is the distance to the ground. The function opens down. The graph of the function does not have a horizontal asymptote.
2. For a sequence: I like eating apples. If I have enough time, I will eat apples as my breakfast. I can recall the number of apples I ate every morning in last week. The sequence of the numbers is 2, 0, 1, 0, 1, 1, 0. Also, I do not make a plan of how many apples I will eat for next week. Therefore, this sequence is diverges.
3. For a series: Every UBC student has own bank card. In my saving account, there is some interests for me. To assume A represents for my final amount, P as my initial amount, r as the rate of yearly interest, n as number of times yearly interest is compounded in one year, t as time. My series can be A=P(1+r/n)^nt.

Everything is so hard. Not only of solving problems, also some technical  problems, such as typing mathematical symbols.