The intercepts made by the tangent to the curve `y=int_(0)^(x) |t| dt,` which is parallel to the line `y=2x` on y-axis are equal to

Options

A. 1, -1
B. `-2, 2`
C. 3
D. -3

Answer

To solve the problem, we need to find the intercepts made by the tangent to the curve \( y = \int_{0}^{x} |t| \, dt \) that is parallel to the line \( y = 2x \) on the y-axis. Here’s a step-by-step solution:

### Step 1: Understand the curve
The curve is defined by the integral:
\[ y = \int_{0}^{x} |t| \, dt \]
We need to evaluate this integral based on the value of \( x \).

### Step 2: Evaluate the integral
For \( x \geq 0 \):
\[ y = \int_{0}^{x} t \, dt = \left[ \frac{t^2}{2} \right]_{0}^{x} = \frac{x^2}{2} \]

For \( x < 0 \):
\[ y = \int_{0}^{x} -t \, dt = -\left[ \frac{t^2}{2} \right]_{0}^{x} = -\frac{x^2}{2} \]

Thus, we can write:
\[ y = \begin{cases} \frac{x^2}{2} & \text{if } x \geq 0 \\ -\frac{x^2}{2} & \text{if } x < 0 \end{cases} \]

### Step 3: Differentiate the curve
To find the slope of the tangent, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

### Step 4: Find the slope for the tangent
We want the tangent to be parallel to the line \( y = 2x \), which has a slope of 2. Therefore, we set:
\[ \frac{dy}{dx} = 2 \]

### Step 5: Solve for \( x \)
For \( x \geq 0 \):
\[ x = 2 \]

For \( x < 0 \):
\[ -x = 2 \implies x = -2 \]

### Step 6: Find the corresponding \( y \) values
Now we will find the \( y \) values for \( x = 2 \) and \( x = -2 \).

1. For \( x = 2 \):
\[ y = \frac{2^2}{2} = \frac{4}{2} = 2 \]

2. For \( x = -2 \):
\[ y = -\frac{(-2)^2}{2} = -\frac{4}{2} = -2 \]

### Step 7: Find the equation of the tangent lines
The tangent line at \( (2, 2) \) with slope 2 is given by:
\[ y - 2 = 2(x - 2) \implies y = 2x - 2 \]

The tangent line at \( (-2, -2) \) with slope 2 is given by:
\[ y + 2 = 2(x + 2) \implies y = 2x + 2 \]

### Step 8: Find the y-intercepts
1. For the tangent line \( y = 2x - 2 \):
- Setting \( x = 0 \):
\[ y = 2(0) - 2 = -2 \]

2. For the tangent line \( y = 2x + 2 \):
- Setting \( x = 0 \):
\[ y = 2(0) + 2 = 2 \]

### Final Result
The intercepts made by the tangent lines on the y-axis are \( -2 \) and \( 2 \).