The slope at any point of a curve y=f(x) is given by `(dy)/(dx)= 2x` and it passes through `(1,-1)`. The equation of the curve is

Options

A. `y=x^(2) + 2`
B. `y= -x^(2) + 2`
C. `y=x^(2) -2`
D. `y= -x^(3)-2`

Answer

To find the equation of the curve given that the slope at any point is \(\frac{dy}{dx} = 2x\) and it passes through the point \((1, -1)\), we can follow these steps:

### Step 1: Write the differential equation
We start with the given slope:
\[ \frac{dy}{dx} = 2x \]

### Step 2: Separate the variables
We can rewrite this as:
\[ dy = 2x \, dx \]

### Step 3: Integrate both sides
Now we integrate both sides:
\[ \int dy = \int 2x \, dx \]
The left side integrates to \(y\), and the right side integrates to \(x^2 + C\):
\[ y = x^2 + C \]

### Step 4: Use the initial condition to find \(C\)
We know the curve passes through the point \((1, -1)\). We can substitute \(x = 1\) and \(y = -1\) into the equation to find \(C\):
\[ -1 = (1)^2 + C \]
\[ -1 = 1 + C \]
\[ C = -1 - 1 = -2 \]

### Step 5: Write the final equation of the curve
Now we substitute \(C\) back into the equation:
\[ y = x^2 - 2 \]

### Final Answer
The equation of the curve is:
\[ y = x^2 - 2 \]

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