Exam tomorrow need urgent help

Question

Q1) Find f'(x) if f(x) = ∫ √(t^4 + 1) dt  between the interval tan (x) and 1

 

Q2) Calculate dy/dx, leave answer in terms of x and y

y tan(x+y) = yx^2

 

Q3) John found he has 18 coins in his piggy bank (containing 10c, 20c and 50c coins) with total value $3. What combinations of 10c, 20c and 50c coins could john have?

Answer to this is (6,12,0) , (9,8,1), (12,4,2), (15,0,3)

Answer 1

y tan(x+y) = yx²

tan(x+y) = x²

 (d /dx)(tan(x+y) =  (d /dx) (x² ) 

sec²(x+y)(1 + dy/dx)= 2x

sec²(x+y) + sec²(x+y) dy/dx = 2x

sec²(x+y) dy/dx = 2x - sec²(x+y)

dy/dx = [ 2x - sec²(x+y)] / sec²(x+y)

         = [ 2x - (tan²(x+y) +1 )] / (tan²(x+y) +1)

         =[ 2x - ((x²)² +1 )] / ((x²)² +1)

            =[ 2x - x⁴ - 1] / (x⁴ +1)

Answer 2

(3)

Given 18 coins that are 10c , 20c , 50c .

All the coins have make sum of $3 = 300c.

To write the all possible combinations we had make the following assumptions .

In this question we don't have any restriction regarding that there must be at least one of each type of coin.

Using these  facts, you can search for solutions for problem as follows: 

(1) Arbitrarily choose the numbers of 50c and 20c you want to try.

(2) For that combination of 50c and 20c , 

Comments

Contd ...

  consider all the possible numbers of 10c you could have(means sum of the all coins is 18).

Number of 50c	Number of 20c	Number of 10c	Total amount	Status of condition
0	5	13	230c	No
0	6	12	240c	No
0	8	10	260c	No
0	10	8	280c	No
0	12	6	300c	Yes
0	14	4	320c	No
1	5	12	270c	No
1	6	11	290c	No
1	8	9	300c	Yes
1	9	8	310c	No
2	2	14	280c	No
2	3	13	290c	No
2	4	12	300c	Yes
2	5	11	310c	No
3	0	15	300c	Yes
3	1	14	310c	No

When no 50c and number of 20c are 5 then we have to take remaining 13 number of 10c to satisfy total 18 coin .

But the total leads to 230c . so now increase the number of 20c 's and repeat it .

So the possiable number of combination of 10c , 20c , 50c are (6,12,0) , (9,8,1) , (12,4,2) and (15,0,3) .