The tangent to y = (ax)+b/(sqrt x) at x =1 is 2x-y=1. Find a and b.
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To begin solving this question, we need to find the slope of the tangent line at x = 1. The slope of the tangent line can be found using calculus by taking the derivative of the function y = (ax+b)/√x. Here's how to find it step by step:
Step 1: Differentiate the function with respect to x:
To differentiate (ax + b) / √x, we need to use the quotient rule.
The derivative of (ax + b) is a.
The derivative of √x is (1 / (2√x)).
Using the quotient rule, we have:
dy/dx = [(a * 2√x) - ((ax + b) * (1 / (2√x))))] / (√x)^2
Simplifying this expression:
dy/dx = (2a√x - (ax + b) / (2√x)) / x
Step 2: Substitute x = 1 into the derivative equation:
To find the slope of the tangent line at x = 1, we need to substitute x = 1 into the derivative equation we found in step 1.
dy/dx = (2a√1 - (a(1) + b) / (2√1)) / 1
dy/dx = (2a - (a + b)) / 2
Step 3: Set the derivative equal to the slope given (2x - y = 1):
Since the derivative represents the slope of the tangent line, we can set it equal to the slope given in the question, which is 2x - y = 1.
(2a - (a + b)) / 2 = 2
Step 4: Solve for a and b:
To find the values of a and b, we can now solve the equation derived in step 3.
Multiplying both sides of the equation by 2 to eliminate the fraction gives:
2a - (a + b) = 4
Simplifying the equation:
a - b = 4
This equation represents a relationship between a and b. There are infinitely many possible values of a and b that satisfy this equation.
So, we cannot determine unique values for a and b based on the information given. The values of a and b can be any pair of numbers that satisfy the equation a - b = 4.