least one of the answers above is NOT correct.
ind the equation of the tangent line to the curve at the point . The equation of this tangent line can be written in
he form . Find and .

ote: You can earn partial credit on this problem.

Ask by Riley Love. in the United States

Jan 19,2025

Question

least one of the answers above is NOT correct.
ind the equation of the tangent line to the curve at the point . The equation of this tangent line can be written in
he form . Find and .

ote: You can earn partial credit on this problem.

Ask by Riley Love. in the United States

Jan 19,2025

UpStudy AI · Accepted Answer

To find the equation of the tangent line to the curve $$ y = 15x e^{x} $$ at the point $$ (0, 0) $$, we need to follow these steps:

1. **Find the derivative of the function** to determine the slope of the tangent line at the given point.
2. **Evaluate the derivative at the point $$ x = 0 $$** to find the slope $$ m $$.
3. **Use the point-slope form of the line** to write the equation of the tangent line.

### Step 1: Find the derivative

The function is given by:
$$
y = 15x e^{x}
$$

To find the derivative $$ \frac{dy}{dx} $$, we will use the product rule, which states that if $$ u = 15x $$ and $$ v = e^{x} $$, then:
$$
\frac{dy}{dx} = u'v + uv'
$$
where $$ u' = 15 $$ and $$ v' = e^{x} $$.

### Step 2: Calculate the derivative

Now, applying the product rule:
$$
\frac{dy}{dx} = 15 e^{x} + 15x e^{x}
$$

### Step 3: Evaluate the derivative at $$ x = 0 $$

Now we will evaluate the derivative at $$ x = 0 $$:
$$
\frac{dy}{dx} \bigg|_{x=0} = 15 e^{0} + 15(0)e^{0} = 15 \cdot 1 + 0 = 15
$$

Thus, the slope $$ m $$ of the tangent line at the point $$ (0, 0) $$ is:
$$
m = 15
$$

### Step 4: Use the point-slope form

The point-slope form of the line is given by:
$$
y - y_1 = m(x - x_1)
$$
Substituting $$ (x_1, y_1) = (0, 0) $$ and $$ m = 15 $$:
$$
y - 0 = 15(x - 0)
$$
This simplifies to:
$$
y = 15x
$$

### Conclusion

In the form $$ y = mx + b $$:
- The slope $$ m = 15 $$
- The y-intercept $$ b = 0 $$

Thus, the correct values are:
$$
m = 15, \quad b = 0
$$

The provided values $$ m = 15 e^{2} $$ and $$ b = 0 $$ are incorrect. The correct answer is $$ m = 15 $$ and $$ b = 0 $$.
$$ m = 15 $$ and $$ b = 0 $$.

least one of the answers above is NOT correct.
ind the equation of the tangent line to the curve at the point . The equation of this tangent line can be written in
he form . Find and .

ote: You can earn partial credit on this problem.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Calculus Questions

least one of the answers above is NOT correct. ind the equation of the tangent line to the curve at the point . The equation of this tangent line can be written in he form . Find and . ote: You can earn partial credit on this problem.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Calculus Questions

least one of the answers above is NOT correct.
ind the equation of the tangent line to the curve at the point . The equation of this tangent line can be written in
he form . Find and .

ote: You can earn partial credit on this problem.