Find the smallest positive integer n such that sqrtx-sqrt(x-1)<0.01
sqrt x = sqrt(x-1) + 0.01
x = (x-1) + 0.02 sqrt(x-1) + 0.0001
0.02 sqrt (x-1) = 0.9999
sqrt (x-1) = 49.995
x-1 = 2499.5
x = 2500.5
To solve this inequality, we can use algebraic manipulations.
Given the inequality:
√x - √(x - 1) < 0.01
We can start by isolating one of the square roots. We'll move √(x - 1) to the right side:
√x < 0.01 + √(x - 1)
Next, we'll square both sides of the inequality to eliminate the square roots:
(√x)^2 < (0.01 + √(x - 1))^2
Simplifying the right side:
x < 0.0001 + 0.02√(x - 1) + (x - 1)
Combine like terms:
0 < 0.02√(x - 1)
To eliminate the square root, we'll square both sides again:
(0)^2 < (0.02√(x - 1))^2
Simplifying:
0 < 0.0004(x - 1)
0 < 0.0004x - 0.0004
Adding 0.0004 to both sides:
0.0004 < 0.0004x
Dividing both sides by 0.0004:
1 < x
Therefore, the smallest positive integer n that satisfies the inequality is n = 2.
To find the smallest positive integer n that satisfies the inequality sqrt(x) - sqrt(x-1) < 0.01, we need to solve the inequality algebraically.
Let's start by isolating the square root term:
sqrt(x) - sqrt(x-1) < 0.01
Now, we can multiply both sides of the inequality by sqrt(x) + sqrt(x-1) to eliminate the radicals in the equation:
(sqrt(x) - sqrt(x-1))(sqrt(x) + sqrt(x-1)) < 0.01(sqrt(x) + sqrt(x-1))
Expanding the left side of the inequality using the difference of squares:
[(sqrt(x))^2 - (sqrt(x-1))^2] < 0.01(sqrt(x) + sqrt(x-1))
Simplifying further:
[x - (x-1)] < 0.01(sqrt(x) + sqrt(x-1))
x - x + 1 < 0.01(sqrt(x) + sqrt(x-1))
1 < 0.01(sqrt(x) + sqrt(x-1))
Next, we can divide both sides of the inequality by 0.01 to isolate the square root terms:
1/0.01 < sqrt(x) + sqrt(x-1)
100 < sqrt(x) + sqrt(x-1)
Now, we square both sides of the inequality to remove the square root terms:
(100)^2 < (sqrt(x) + sqrt(x-1))^2
10000 < x + 2sqrt(x)sqrt(x-1) + (x-1)
Simplifying:
10000 < 2x + 2sqrt(x^2 - x)
We're looking for the smallest positive integer n, so we can start by plugging in n = 1 and check if it satisfies the inequality:
10000 < 2(1) + 2sqrt(1^2 - 1)
10000 < 2 + 2sqrt(0)
10000 < 2
Since 10000 is not less than 2, n = 1 does not satisfy the inequality.
We continue this process, incrementing n until we find the smallest positive integer that satisfies the inequality.