Given f"(x)=5x+2 and f'(-3)=3 and f(-3)=-6
find f'(x)=
find f(3)=
from f"(x)=5x+2
f ' (x) = (5/2)x^2 + 2x + c
but f ' (-3) = 3
3 = (5/2)(9) + 6 + c
c = ...
find c and
now you have f ' (x)
integrate once more, adding another constant
and since you have (-3,6) on it, you can find that constant as well
So once I solve for c, how does that translate to f'(x)?
To find the first derivative, f'(x), we can integrate the second derivative, f"(x). Here's how:
Step 1: Integrate f"(x) to get f'(x).
When you integrate 5x+2, you get (5/2)x^2 + 2x + C. The constant of integration, C, represents the unknown constant in the antiderivative, and it can take any value.
So, f'(x) = (5/2)x^2 + 2x + C, where C is an unknown constant.
Step 2: Find the value of the constant C using the given information.
We are given that f'(-3) = 3. Substituting -3 for x in f'(x) and setting it equal to 3, we have:
(5/2)(-3)^2 + 2(-3) + C = 3
Simplifying, we get:
(5/2)(9) - 6 + C = 3
45/2 - 6 + C = 3
45/2 - 12/2 + C = 3
(45 - 12)/2 + C = 3
33/2 + C = 3
Now, we subtract 33/2 from both sides to isolate C:
C = 3 - 33/2
C = (6 - 33)/2
C = -27/2
So, the constant C is -27/2.
Therefore, f'(x) = (5/2)x^2 + 2x - 27/2.
To find f(3), we can integrate f'(x) or use the given information.
Step 1: Integrate f'(x) to get f(x).
When you integrate (5/2)x^2 + 2x - 27/2, you get (5/6)x^3 + x^2 - (27/2)x + C. Again, C represents the constant of integration.
So, f(x) = (5/6)x^3 + x^2 - (27/2)x + C, where C is an unknown constant.
Step 2: Find the value of the constant C using the given information.
We are given that f(-3) = -6. Substituting -3 for x in f(x) and setting it equal to -6, we have:
(5/6)(-3)^3 + (-3)^2 - (27/2)(-3) + C = -6
Simplifying, we get:
(5/6)(-27) + 9 + (81/2) + C = -6
(-5/2)(9) + 9 + (81/2) + C = -6
(-45/2 + 9 + 81/2) + C = -6
(-45/2 + 18/2 + 81/2) + C = -6
(54/2) + C = -6
27 + C = -6
Now, subtract 27 from both sides to isolate C:
C = -6 - 27
C = -33
So, the constant C is -33.
Therefore, f(x) = (5/6)x^3 + x^2 - (27/2)x - 33.
Finally, to find f(3), substitute 3 for x in f(x):
f(3) = (5/6)(3)^3 + (3)^2 - (27/2)(3) - 33
Simplifying, we get:
f(3) = (5/6)(27) + 9 - (81/2) - 33
f(3) = 45/2 + 9 - 81/2 - 33
f(3) = (45 + 18 - 81 - 66)/2
f(3) = -84/2
f(3) = -42