The graph of
f(x)
passes through the point
(0,3)
. The slope
of
f
at any point
P
is 3 times the
y-
coordinate of
P
.
Find
f(1)
To find the value of f(1), we need to determine the equation of the function f(x) using the information given.
Let's start by finding the equation of the function f(x) using the point (0,3).
Since the slope of f at any point P is 3 times the y-coordinate of P, and the point (0,3) lies on the graph, we can find the equation of the tangent line at that point.
The equation of a line can be written as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In this case, we have (x1, y1) = (0, 3) and m = 3 * y-coordinate = 3 * 3 = 9.
Plugging in these values, we get y - 3 = 9(x - 0), which simplifies to y - 3 = 9x.
Now, we can find f(x) by integrating the equation of the tangent line. Integrating y - 3 = 9x with respect to x, we get y = 9x^2/2 + C, where C is the constant of integration.
To find the value of C, we can use the fact that the point (0,3) lies on the graph of f(x). Substituting x = 0 and y = 3 into the equation y = 9x^2/2 + C, we get 3 = C.
Therefore, the equation of the function f(x) is y = 9x^2/2 + 3.
Finally, we can find f(1) by plugging in x = 1 into the equation y = 9x^2/2 + 3:
f(1) = 9(1)^2/2 + 3
= 9/2 + 3
= 9/2 + 6/2
= 15/2
So, f(1) = 15/2.
To find f(1), we need to determine the equation of the function f(x) first.
Since the slope of the function at any point P is 3 times the y-coordinate of P, we can write the equation of the slope at point P as:
slope = 3 * y-coordinate
Now, let's use the given point (0,3) to find the equation:
slope = 3 * 3 = 9
Therefore, the slope of the function at any point is 9.
Now, let's find the equation of the function f(x) using the point-slope form:
y - y1 = m(x - x1)
Substituting the value of the point (0,3) and the slope 9:
y - 3 = 9(x - 0)
y - 3 = 9x
Simplifying the equation, we get:
y = 9x + 3
Now, we can find f(1) by substituting x = 1 into the equation:
f(1) = 9(1) + 3
f(1) = 9 + 3
f(1) = 12
Therefore, f(1) = 12.