Find the value of c where the line l tangent to the graph of y=2.5*5^x at (0,2.5) intersects the x-axis
-1/ln5
Yup, that is the answer I had given you when you posted this before.
Why are you reposting it ?
http://www.jiskha.com/display.cgi?id=1349142347
To find the value of c, we need to find the equation of the tangent line l. Let's start by finding the derivative of the function y = 2.5 * 5^x.
Step 1: Find the derivative of y with respect to x:
To find the derivative of y = 2.5 * 5^x, we can use the power rule for differentiation.
dy/dx = (ln(5)) * 2.5 * 5^x
Step 2: Find the slope of the tangent line at (0, 2.5):
Since the tangent line is tangent to the graph at (0, 2.5), we can use the derivative at that point to find the slope of the tangent line.
slope = dy/dx | (x=0) = (ln(5)) * 2.5 * 5^0 = (ln(5)) * 2.5
Step 3: Find the equation of the tangent line:
Using the slope-point form of a line, we can write the equation of the tangent line l:
y - y1 = m(x - x1)
where (x1, y1) is the point of tangency.
Substituting the values into the equation, we get:
y - 2.5 = (ln(5)) * 2.5 * (x - 0)
Simplifying further, we have:
y - 2.5 = (ln(5)) * 2.5x
y = (ln(5)) * 2.5x + 2.5
Step 4: Find the point of intersection with the x-axis:
To find the point of intersection with the x-axis, we set y = 0 and solve for x.
0 = (ln(5)) * 2.5x + 2.5
Subtracting 2.5 from both sides, we get:
(ln(5)) * 2.5x = -2.5
Dividing both sides by (ln(5)) * 2.5, we have:
x = -2.5 / ((ln(5)) * 2.5)
Simplifying further, we get:
x = -1 / ln(5)
Therefore, the value of c is -1 / ln(5).