Solution to the problem of angle proving in junior high school geometry In the right triangle ABC, the angle ACB = 90 degrees, AC = BC, P is a point in the triangle, and Pb = 1, PC = 2, PA = 3. Find the degree of angle BPC

Solution to the problem of angle proving in junior high school geometry In the right triangle ABC, the angle ACB = 90 degrees, AC = BC, P is a point in the triangle, and Pb = 1, PC = 2, PA = 3. Find the degree of angle BPC

135 degrees
You can rotate the triangle APC 90 degrees clockwise around point C, and prove that it is the sum of 90 degrees and 45 degrees by using congruence and Pythagorean theorem

Derivative formula of variable upper limit integral
The upper limit of function ∫ x * f (x) DX is Q and the lower limit is 0?

Let's push it by definition. Suppose ∫ XF (x) DX = f (x), then f '(x) = XF (x)
Then ∫ (0, q) XF (x) DX
= F(Q)-F(0)
The result is f '(q) = QF (q)

Derivation of variable upper bound integral
Please write down the steps,

The integral variable is t, so you can take x out of the integral
 

Derivative formula of variable upper limit integral
Q: if f (x) = ∫ (upper limit x, lower limit a) XF (T) DT, then f '(x) =?
The answer is this:
X is not an integral variable
F (x) = x ∫ (upper limit x, lower limit a) f (T) DT
Then f '(x) = (upper limit x, lower limit a) f (T) DT + XF (x)
What I can't understand is: in the final answer, why don't we subtract another AF (a), because there are upper and lower limits for the integral. When we seek the derivative, don't we also need the derivative of the upper limit minus the derivative of the lower limit?

F(x) = ∫(a,x) xf(t) dt
F(x) = x∫(a,x) f(t) dt
F'(x) = ∫(a,x) f(t) dt + x * [x' * f(x) - a' * f(a)]
=(1 / x) f (x) + X * [1 * f (x) - 0 * f (a)], the derivative of lower bound a is not zero, so the whole will be zero
= (1/x)F(x) + xf(x)

Proof of derivative formula of variable limit integral

The upper limit is a (x), and the lower limit is B (x) y = (a (x), B (x)) ∫ f (T) DT. It is known that the original function of F (x) is f (x), f '(x) = f (x) (observe y = (a, b) ∫ f (T) DT = f (a) - f (b), and substitute it in the brackets) so y = (a (x), B (x)) ∫ f (T) DT = f [a (x)] - f [b (x)] and take the derivation on both sides of Y' = (f [a (x)] '- (f [b (x)])'

What is the formula for calculating the final value and present value of an ordinary annuity?
It's better to give examples

FV = PMT * [(1 + I) + (1 + I) ^ 2 + (1 + I) ^ 3 +... + (1 + I) ^ n] final value
PV = PMT * [(1 + I) ^ (- 1) + (1 + I) ^ (- 2) + (1 + I) ^ (- 3) +... + (1 + I) ^ (- n)] present value
PMT is the amount of each payment. The above formula assumes that the amount of each payment is constant and equal

What are annuity, ordinary annuity, present value and final value of ordinary annuity? What is the difference between them?

Let's put it bluntly: annuity: the same amount occurs at the same time, if not at the same time and amount, it will not be counted. For example, if it occurs at the end of the month, the amount of each subsequent period also occurs at the end of the month, the end of the quarter, the end of the year, or how many days, it is the same. Ordinary annuity: the amount that occurs at the end of each period

What is the formula for calculating the present value of an immediate annuity?
P=A·［（P/A,i,n-1）+1］
For example: an enterprise rents an equipment and pays 5000 yuan at the beginning of each year in 10 years, with an annual interest rate of 8%. Q: what is the present value of these rents?
Known: a = 5000 I = 8% n = 10 find: P
P=A×[（P/A,i,n-1）+1]
=5000×[（P/A,8%,9）+1]
=5000 × (6.247 + 1) = 36235 yuan
In this question, I want to know how to get its 6.247?
In other words, (P / A, 8%, 9) what does the comma mean? What is the relationship between the three data? Multiplication or division? How to calculate?

(P / A, 8%, 9) refers to the present value coefficient of ordinary annuity with 8% discount rate and 9 periods. (P / A, 8%, 9) can be found by looking up the "table of present value coefficient of annuity", that is, the corresponding value of 8% interest rate and 9 periods on the table. The value is 6.247
I don't know if I make myself clear

How to deduce the formula of the present value of the post paid annuity, the final value of the post paid annuity, the present value of the pre paid annuity and the final value of the pre paid annuity

The deduction formula of the present value of the post paid annuity is calculated according to the compound interest present value method. The formula of the present value of the annuity is: P = a (1 + I) ^ - 1 + a (1 + I) ^ - 2 + a (1 + I) ^ - 3 + +A (1 + I) ^ - N multiply both sides by (1 + I) to get: P (1 + I) = a (1 + I) + a (1 + I) ^ - 1 + a (1 + I) ^ - 2 + +A (1 + I) ^ - (n-1) P = a * {[1 - (1 +...)

What is the formula for calculating the present value of an advance annuity

P/A=P*(P/A,n.i)*(1+i)