find the equation of the line tangent to the graph y=3e^x at x=1
Well, let's put on our math hats and figure this out!
Now, to find the equation of the tangent line, we need two things: the slope of the tangent line and a point that lies on the line.
To get the slope of the tangent line, we can find the derivative of the function y = 3e^x. The derivative of e^x is just e^x, so the derivative of y = 3e^x is dy/dx = 3e^x.
Now that we have the slope, we need a point on the line. Since we're looking for the tangent at x = 1, we can substitute this value back into the original equation to find the corresponding y-value. Plugging in x = 1, we get y = 3e^1 = 3e.
So, we know that the point (1, 3e) lies on the tangent line.
Using the point-slope form of a line, y - y₁ = m(x - x₁), we can plug in the slope (3e^x) and the point (1, 3e):
y - 3e = 3e^1(x - 1)
Simplifying a bit, we get:
y - 3e = 3ex - 3e
And rearranging to get the equation into slope-intercept form (y = mx + b), we find:
y = 3ex - 3e + 3e
Which simplifies to:
y = 3ex
So, the equation of the tangent line to the graph y = 3e^x at x = 1 is y = 3ex.
And with that, we've nailed the math and had a little fun along the way!
To find the equation of the line tangent to the graph of y = 3e^x at x = 1, we can follow these steps:
Step 1: Find the derivative of the function y = 3e^x.
Step 2: Evaluate the derivative at x = 1 to find the slope of the tangent line.
Step 3: Use the slope found in step 2 and the point (1, y) on the graph to find the equation of the tangent line using the point-slope form.
Let's start with step 1:
The function y = 3e^x is in the form y = ae^x, where a is a constant. The derivative of y = ae^x with respect to x is simply a * e^x.
In our given function, the constant a is 3, so the derivative of y = 3e^x is:
dy/dx = 3 * e^x
Now, let's move to step 2:
We need to evaluate the derivative at x = 1. Substitute x = 1 into the derivative function:
dy/dx = 3 * e^1
dy/dx = 3e
So, the slope of the tangent line at x = 1 is 3e.
Moving on to step 3:
We have the slope of the tangent line (3e) and a point on the graph (x = 1, y = 3e^1 = 3e).
Using the point-slope form of a line, we can write the equation of the tangent line:
y - y1 = m(x - x1)
Where m is the slope and (x1, y1) is the point on the graph.
Plugging in the values, we get:
y - (3e) = (3e)(x - 1)
Simplifying further:
y - 3e = 3ex - 3e
Finally, rearranging the equation to the usual slope-intercept form:
y = 3ex - 3e + 3e
Which simplifies to:
y = 3ex
So, the equation of the line tangent to the graph of y = 3e^x at x = 1 is y = 3ex.