Question

Find a curve with a positive derivative through the point (1,1) whose length integral is given below. b. How many such curves are there? Give reasons for your answer. 4 L= 1 + dx 4x 2 a. Let the curve be y=f(x). Determine dy dx 2 dx dy Determine dx dy = + dx Because the curve is to have a positive derivative, an equation of the curve is y= As x increases from the lower limit to the upper limit, the curve runs from to (Type ordered pairs.) b. How many such curves are there? Explain. Choose the correct answer below. A. None. The function from the previous step is not defined at the given x-value. B. Infinitely many. Any arbitrary constant can be added to the function from the previous step. O C. Only one. The function from the previous step is a constant function. OD. Only one. The derivative of the function is known, as well as the value of the function at one value of x.

Answer 1

To find a curve with a positive derivative through the point (1,1) whose length integral is given, we integrate the derivative of y=f(x) and solve for the equation of the curve. The equation of the curve is y = -ln(x) + 1/(8x).

Step-by-step explanation:

To find a curve with a positive derivative through the point (1,1) whose length integral is given, we first simplify the integral expression: ∫[1+dx/(4x^2)] dx. This simplifies to ∫(1/x^2 + 1/(4x^3)) dx. The antiderivative of 1/x^2 is -1/x and the antiderivative of 1/(4x^3) is -1/(8x^2). So the integral becomes -1/x - 1/(8x^2).

To find the equation of the curve, we integrate the derivative of y=f(x). The derivative is dy/dx = -1/x - 1/(8x^2). Integrating this, we get y = -ln(x) + 1/(8x) + C, where C is a constant. To determine the curve that passes through the point (1,1), we substitute x=1 and y=1 into the equation and solve for C. After substituting these values, we get 1 = -ln(1) + 1/8 + C, which simplifies to C = 0.

Therefore, the equation of the curve is y = -ln(x) + 1/(8x). As x increases from the lower limit to the upper limit, the curve runs from (1,1) to (x, y).