The line tangent to the graph y=3e^x at x=3

Answer 1

To find a tangent line, we need to start off by finding the tangent point.

To do this, we evaluate the function and determine what y is equal to when x = 3.

Plugging into our equation, y = 3e³ when x=3.

We can now find the slope of y = 3e^x by taking the derivative of the equation.

The derivative of our equation: dy/dx = 3e^x .

By definition, the derivative of our equation presents us with the slope which is m=3e^x, which becomes m=3e³ when x=3.

Now, we need to find the line with slope we just found through the tangent point (3, 3e³).

As a reminder, our line formula is y=mx+b.

So far, we have y = 3e³*x+b.

We need to find the appropriate b (or y-intercept) for our tangent line. Let's use our tangent point.

3e³⁼3e³*(3)+b We have this by plugging in our tangent point and the slope which we found earlier.

Isolating b, we have b= -6e³.

To complete our tangent line, we have y=3e³x-6e³.

Answer 2

To find the tangent line at x=3, you can first find the derivative at x=3. This will give you the slope

Take the derivative:

y' = 3e^x

At x = 3, y' = 3e^3 = 60.26

This means the tangent line has the form y = 3e^3 x + b

The tangent line will also touch the graph at x=3. At x=3, y = 3e^3 = 60.26

Tangent line:

y = 3e^3 x + 3e^3

or

y = 60.26x + 60.26