Differentiable

When we say that a function is differentiable in an interval, it means that this function’s derivative exists at each point in this interval.

When we say derivative exists at certain point a, it means we can pick a new point a+c and use function at a+c minus function at a, then divided by c.

When a+c is approaching a, which means c is sufficiently small, if the limit of above equation still exists, then we can say it is derivative at this point.

If a function can be derivative at every point in a interval, we can say this function is differentiable in this interval.

When an function is analytic at c, it means that this function can be differentiable infinitely and it is converges to f(x) for x near c. This definition is useful because it can change a function into a infinite power series. This is more convenient for us to figure out many problems, such as whether this function converges or estimate decimal result.

A function differentiable infinitely does not mean it is analytic, such as problem.

However, an analytic function can be differentiable infinitely.

After I review my midterm exam, I found that I lose many points on the last step of integration. Every time when I finished the simplification part and I calculate the antiderivative of  the function I get a wrong answer for the miscalculation. For example, in the (f) question of first part, I simplified it into one over u to the power three and its antiderivative should be minus one over two times one over u to the power two but I got four instead, which leads to a wrong answer.

From the midterm, I noticed that this kind of mistakes are the main factor of losing mark. In order to solve this kind of problem, I should do more practice on antiderivative and develop a sense to those function, which can help me figure problems quickly during the final exam. In addition, I should get familiar with the antiderivative of the common functions, such as power function or trigonometric function. 

Tips

1. Before you doing a substitution, observe which part of the function is the derivative of the other part of this function.
2. If the function is not so clear that you cannot find out the substitution immediately. Make some tries instead of doing nothing and you may find idea during this process.
3. Remember the derivative of those important function, such as sin or cos, to help you find out the substitution faster and it can also save your time in a exam.

According to the definition of integrable, we divide the area into infinite rectangle and each rectangle have same width. Then, we pick up a f(t*) in each subinterval as the height of each rectangle and calculate each rectangle’s area and sum up.

In this case, there are at least 3 subintervals we can make f(t)=0 (at most 6 if t=1, 2, 3 is at the end point of its subinterval).

When the discontinuous point is finite, we can ignore this different of sum because we take the limit as n growing to infinite and there exist another part of sum which is infinite.

A street light on the East Mall is at the top of a 13-foot-tall pole. A 7-foot-tall man walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of his shadow moving when he is 40 feet from the base of the pole?

Here is the solution:

The distance of shadow: Ss

The distance of man: Sp

Considering the relationship of distance between man and his shadow, we get:

Ss/(Sp-Ss) = 13/7

Ss = (13/6)*Sp

We take the derivative of the equation on both side, then we get:

dSs/dt = (13/6)*(dSp/dt)

Because the man’s spend is 5 ft/sec so dSp/dt = 5 and we get dSs/dt = 65/6.

(a)Intermediate Value Theorem is that when a function is continuous, we can pick a number c in a interval (a,b) and there exist f(c) in a interval (f(a), f(b))  . To make someone with no knowledge of the IVT, we can give an example like taking a plane from Australia to China. As is known to all, the earth is a sphere and there is no way for us to take a plane from Australia to China without crossing the equator. No matter how pilots arrange the air line, there exist a point that he has to cross the equator and that is as the same meaning of IVT, which we can definitely find a c in a interval (a,b) and make f(c) in a interval (f(a), f(b)) if this function is continuous.

Therefore, we can know what this question want to express. Since f(0) is smaller than 0 and f(pi) is greater than 0, there exist a point a in (0, pi) such that f(a)=0.

(b)My argument is relied explicitly on the concept of continuity.

1.A parabola of a ball is a function. As is known to all, the ball will rise till its highest point and then fall down. Its kinetic trajectory is a curve and both the starting point and the ending point are equal to zero so there is no horizontal asymptote of this function.

2.Fibonacci sequence is related to golden ratio, which is used for constructing  buildings, such as calculate Proportion of a house’s area to make it more reasonable for human to live.

According to its property, the latter term equals the sum of the former two terms. Since this sequence will increase constantly, it diverges.

3.Geometric series can be used to calculate the value of an annuity.Since interest is counted monthly, using geometric series help people calculate the final value of an annuity more conveniently. People gain money every month so the rate must greater than 100%, which means the ratio is bigger than 1. Therefore, this series will diverge.

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