Find the function f, given that the slope of the tangent line at any point (x,f(x)) is f '(x) and that the graph of f passes through the given point.
f '(x)=6(2x-7)^5 at (4, 3/2)
f=INT f= (2x-7)^6 + C
now find C
3/2 = (2*4-7)^6 + C
C=1/2
if dy/dx = 6(2x-7)^5
y = (6/6) (2x-7)^6 (1/2) + c
y = (1/2)(2x-7)^6 + c
but (4 , 3/2) lies on it, so
3/2 = (1/2)(1)^6 + c
3/2 = 1/2 + c
c = 1
f(x) = (1/2)(2x-7)^6 + 1
To find the function f given the slope of the tangent line at any point and a point on the graph, we need to integrate the derivative f'(x) with respect to x.
Given:
f '(x) = 6(2x-7)^5
Point: (4, 3/2)
Step 1: Integrate f'(x) to find f(x)
To integrate f'(x), we use the power rule of integration:
∫(6(2x-7)^5) dx
= 6 ∫(2x-7)^5 dx
Using the power rule, we can integrate (2x-7)^5:
= 6 * (1/6) * (2x-7)^6 + C
Simplifying further:
= (2x-7)^6 + C
Step 2: Use the given point (4, 3/2) to find the value of the constant C
The graph passes through the point (4, 3/2), which means that when x = 4, f(x) = 3/2.
Substituting the values into the function, we have:
(2x-7)^6 + C = 3/2
Now, substitute x = 4:
(2 * 4 - 7)^6 + C = 3/2
Simplifying:
(8 - 7)^6 + C = 3/2
(1)^6 + C = 3/2
1 + C = 3/2
Subtracting 1 from both sides, we get:
C = 3/2 - 1
C = 1/2
Step 3: Write the final function f(x)
Now that we have the value of the constant C, we can rewrite the function f(x):
f(x) = (2x-7)^6 + 1/2
So, the function f(x) is (2x-7)^6 + 1/2.
To find the function f, we need to integrate the derivative f '(x). In this case, the derivative is f '(x) = 6(2x-7)^5.
To integrate f '(x), we apply the power rule for integration. According to the power rule, the integral of x^n is (1/(n+1)) * x^(n+1). Applying this rule to f '(x), we get:
∫ f '(x) dx = ∫ 6(2x-7)^5 dx
To integrate, we can use a u-substitution. Let u = 2x-7. Then, du = 2 dx. Rearranging this equation, we get dx = (1/2) du.
Substituting these values, we can rewrite the integral:
∫ 6(2x-7)^5 dx = ∫ 6u^5 (1/2) du = 3u^5 du
Now, we can integrate this expression:
∫ 3u^5 du = (3/6) * u^6 + C = (1/2)u^6 + C
Returning to our original variable x, we replace u with 2x-7 in the integrated expression:
f(x) = (1/2)(2x-7)^6 + C
To find the specific value of the constant C, we use the given point (4, 3/2). Plugging these values into the equation, we solve for C:
3/2 = (1/2)(2(4)-7)^6 + C
3/2 = (1/2)(8-7)^6 + C
3/2 = (1/2)(1)^6 + C
3/2 = (1/2)(1) + C
3/2 = 1/2 + C
C = 3/2 - 1/2
C = 2/2
C = 1
Therefore, the function f(x) is:
f(x) = (1/2)(2x-7)^6 + 1