Explanation

The hyperbolic tangent function tanh(y) is defined as:

tanh(y) = (e^y - e^(-y)) / (e^y + e^(-y))

Explanation

Given y = x²/(1 + x²),

∫y dx = ∫[x²/(1 + x²)] dx = ∫[1 - 1/(1 + x²)] dx = x - arctan(x) + C.

Given ∫y dx = C + f, f = x - arctan(x).

x - 1 + 1/(x + 1)

Explanation

1 houre = 3600sec

                 =1/100 * 3600

                   =36 km.

SH31-8-2023

Explanation

If the result is 15, and we subtract 9 from it, we get:

15 - 9 = 6

Since we divided the number by 4, we multiply 6 by 4 to get:

6 × 4 = 24

Explanation

To find the value of c that satisfies the Mean Value Theorem, we need to find the derivative of the function f(x) = x^2 + 2x + 1.

f'(x) = 2x + 2

The Mean Value Theorem states that there exists a point c in the interval [1, 2] such that:

f'(c) = (f(2) - f(1)) / (2 - 1)

First, let's calculate f(2) and f(1):

f(2) = 2^2 + 2(2) + 1 = 9

f(1) = 1^2 + 2(1) + 1 = 4

Now, we can calculate the slope:

(f(2) - f(1)) / (2 - 1) = (9 - 4) / 1 = 5

Now, we set f'(c) equal to 5:

2c + 2 = 5

Solving for c, we get:

2c = 3

c = 3/2

Since c = 3/2 is between 1 and 2, it satisfies the Mean Value Theorem.

Explanation

Since the population growth is exponential, the population P can be modeled as:

P = P0 x (1 + r)^t

where P0 is the initial population, r is the growth rate, and t is the time (in years).

Given that the initial population P0 = 20 and the population grew to 25 after 2 years, we can find the growth rate r:

25 = 20 x (1 + r)^2

Dividing both sides by 20:

1.25 = (1 + r)^2

Taking the square root:

1 + r = √1.25 = 1.125

Subtracting 1 from both sides:

r = 0.125

Now we can write the exponential growth function:

P = 20 x (1 + 0.125)^t

P = 20 x (1.25)^t

Explanation

(1) Solve for x in the second equation.

x - 4y = -1

x - 4y + 4y = -1 + 4y

x = 4y - 1

(2) Replace x with 4y - 1 in the first equation.

3(4y-1) + y = 10

12y - 3 + y = 10

13y - 3 = 10

13y - 3 + 3 = 10 + 3

13y = 13

13y/13 = 13/13

y = 1

**

ND17-1-2023

Explanation

2x=32

Step 1: Express 32 as a power of 2

32=2^5

Step 2: Set the exponents equal

2^x=2^5⇒x=5

 Correct answer: 5

Explanation

To simplify the expression, we need to follow the rules of fractions and combine like terms:

X + 1/7 - X - 1/8

First, subtract X from X, which cancels out:

1/7 - 1/8

To subtract fractions, we need a common denominator, which is 56. So we can rewrite the fractions as:

(8/56) - (7/56)

Now we can subtract: 1/56

So the simplified result is:

1/56

Explanation

Given: p/q = 7

We can rewrite this as: p = 7q

Now, let's evaluate the expression: p+q/p-q

Substitute p = 7q: (7q+q)/(7q-q)

Combine like terms: (8q)/(6q)

Simplify: 8/6

Divide both numerator and denominator by 2: 4/3

So, the correct answer is: 4/3