Determine the equation the tangent line on curve 4e^x on (0,4).
Since derivative of 4e^x =4e^x, then I plugged in 0 for X and the slope I got was 4. Then I wrote the equation and got y=4x +4
dy/dx=4e^x, and at x=0, 4e^0=4
so slope is 4. And your equation is correct.
To determine the equation of the tangent line on the curve 4e^x at the point (0,4), we need to find both the slope and the y-intercept.
First, we find the slope by taking the derivative of the curve 4e^x, which is 4e^x. Plugging in the value x=0 into the derivative, we get:
slope = 4e^0 = 4 * 1 = 4
Thus, the slope of the tangent line at the point (0,4) is 4.
Next, we use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the point (0,4) and m is the slope we found.
Plugging in the values, we have:
y - 4 = 4(x - 0)
This simplifies to:
y - 4 = 4x
To get the equation in slope-intercept form (y = mx + b), we isolate y by adding 4 to both sides:
y = 4x + 4
So, the equation of the tangent line on the curve 4e^x at the point (0,4) is y = 4x + 4.
To determine the equation of the tangent line on the curve 4e^x at the point (0,4), follow these steps:
1. Calculate the derivative of the function 4e^x with respect to x. The derivative of e^x is e^x, so the derivative of 4e^x is 4e^x.
2. Substitute the x-coordinate of the given point (0) into the derivative to find the slope of the tangent line. Plugging in x=0, we get: slope = 4e^0 = 4.
3. Knowing the slope of the tangent line (4), we can use the point-slope form of a linear equation to write the equation of the tangent line. The point-slope form is: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Plugging in the values of (x1, y1) = (0, 4) and m = 4, the equation becomes: y - 4 = 4(x - 0).
4. Simplify the equation. Distribute on the right side to get: y - 4 = 4x.
5. Finally, rearrange the equation to the standard form y = mx + b, where m is the slope and b is the y-intercept. The equation is: y = 4x + 4.
Therefore, the equation of the tangent line on the curve 4e^x at the point (0,4) is y = 4x + 4.