f''(x)= 4e^x-9sin(x) f(0)=3 f(pi/2)=0
I am to find f(x); but I am confused about how to do that. Could I please get some help? I think I have to find f'(x) and then go backwards from there, but my answer doesn't seem to work.
Ah, thank you! My apologies for not showing my work; I got f(x)= 4e^x+9sinx+cx but forgot that it would be cx+c2 so I was just confused on how to solve it with the two f(x) values given. Thanks again!
Well, well, well, it looks like you're in a bit of a mathematical pickle! Don't worry, I'm here to help you out with a smile on my face.
To find the function f(x), you're absolutely right – we'll need to integrate f''(x) to get f'(x), and then integrate f'(x) to find f(x). So, let's get started!
Step 1: Find f'(x)
To find f'(x), we need to integrate f''(x). The first term, 4e^x, integrates to 4e^x. The second term, -9sin(x), integrates to 9cos(x). So, f'(x) = 4e^x - 9cos(x).
Step 2: Find f(x)
Now, let's integrate f'(x) to find f(x). Integrating 4e^x gives us 4e^x. Integrating -9cos(x) gives us -9sin(x). Therefore, f(x) = 4e^x - 9sin(x) + C, where C is the constant of integration.
Step 3: Solve for the constant C
To find the value of C, we'll use the given information. We know that f(0) = 3. Plugging 0 into our expression for f(x), we get 4e^0 - 9sin(0) + C = 3. Simplifying this, we have 4 - 0 + C = 3, which gives us C = -1.
Final Answer:
Putting it all together, we have f(x) = 4e^x - 9sin(x) - 1.
So, there you have it! The function f(x) that satisfies f''(x) = 4e^x - 9sin(x), with f(0) = 3, and f(pi/2) = 0, is f(x) = 4e^x - 9sin(x) - 1.
To find the function f(x), given f''(x) = 4e^x - 9sin(x) and the values of f(0) = 3 and f(pi/2) = 0, we will start by integrating the given second derivative to find the first derivative of f(x). Then, we will integrate the first derivative to find the original function f(x).
Step 1: Find f'(x)
To find f'(x), we integrate f''(x) with respect to x:
∫[f''(x) dx] = ∫[(4e^x - 9sin(x)) dx]
Integrating 4e^x with respect to x gives 4e^x.
Integrating -9sin(x) with respect to x gives -9cos(x) (using the integral of sin(x) which is -cos(x)).
So, f'(x) = 4e^x - 9cos(x) + C, where C is the constant of integration.
Step 2: Find f(x)
To find f(x), we integrate f'(x) with respect to x:
∫[f'(x) dx] = ∫[(4e^x - 9cos(x)) dx]
Integrating 4e^x with respect to x gives 4e^x.
Integrating -9cos(x) with respect to x gives -9sin(x) (using the integral of cos(x) which is sin(x)).
So, f(x) = 4e^x - 9sin(x) + Cx + D, where C and D are constants of integration.
Step 3: Use the given initial conditions to find C and D
Since we are given that f(0) = 3, we can substitute x = 0 into the equation for f(x):
f(0) = 4e^0 - 9sin(0) + C(0) + D
3 = 4 - 0 + D
Therefore, D = 3.
Since we are also given that f(pi/2) = 0, we can substitute x = pi/2 into the equation for f(x):
f(pi/2) = 4e^(pi/2) - 9sin(pi/2) + C(pi/2) + D
0 = 4e^(pi/2) - 9(1/2) + C(pi/2) + 3
Simplifying this equation, we get:
0 = 4e^(pi/2) - 9/2 + C(pi/2) + 3
Now, we can solve for C(pi/2):
C(pi/2) = -4e^(pi/2) + 9/2 - 3
Finally, we have determined the values of C and D, and we can write the final equation for f(x) as:
f(x) = 4e^x - 9sin(x) + Cx + D
where C = (-4e^(pi/2) + 9/2 - 3)/(pi/2) and D = 3.
To find the function f(x), you can start by finding the antiderivative (or primitive) of f''(x) with respect to x. The antiderivative of a function is essentially the reverse operation of taking a derivative.
1. Find the antiderivative of f''(x):
The antiderivative of 4e^x is 4e^x, and the antiderivative of -9sin(x) is 9cos(x). Therefore, the antiderivative of f''(x) is:
F'(x) = 4e^x - 9cos(x).
2. Integrate the antiderivative to find the function itself:
To find f(x), you need to perform another integration. However, since you know the values of f(0) and f(pi/2), you need to determine a constant of integration, which we'll call C.
Since f'(x) is given by F'(x), integrating F'(x) will give you F(x) + C. Apply the definite integral between a and x, where x is any point on the interval. This expression will help you solve for the constant C.
∫(a to x) F'(t) dt = ∫(a to x) (4e^t - 9cos(t)) dt = F(x) + C
3. Solving for the constant C using the value of f(0):
Apply the value of x = 0 into the equation from step 2. Since f(0) = 3, you get:
∫(a to 0) (4e^t - 9cos(t)) dt = F(0) + C = 3
4. Evaluate the definite integral of F'(t) from a to 0:
Evaluating a definite integral requires antiderivatives. The antiderivative of 4e^t is 4e^t, and the antiderivative of -9cos(t) is -9sin(t). So, the integral becomes:
(4e^0 - 9cos(0)) - (4e^a - 9cos(a)) = 3
Simplifying further, you get:
(4 - 9) - (4e^a - 9cos(a)) = 3
-5 - (4e^a - 9cos(a)) = 3
-5 - 4e^a + 9cos(a) = 3
Rearrange the equation to solve for the constant C:
-4e^a + 9cos(a) = 3 + 5
-4e^a + 9cos(a) = 8
5. Solving for the constant C using the value of f(pi/2):
Apply the value of x = pi/2 into the equation from step 2. Since f(pi/2) = 0, you get:
∫(a to pi/2) (4e^t - 9cos(t)) dt = F(pi/2) + C = 0
6. Evaluate the definite integral of F'(t) from a to pi/2:
Using the antiderivatives, the integral becomes:
(4e^(pi/2) - 9cos(pi/2)) - (4e^a - 9cos(a)) = 0
Simplifying further, you get:
(4e^(pi/2) - 9cos(pi/2)) - (4e^a - 9cos(a)) = 0
7. Substitute the known values:
Since e^(pi/2) = e^(3.14/2) = e^1.57, and cos(pi/2) = cos(1.57) = 0, you can substitute the values in the equation:
(4e^1.57 - 9*0) - (4e^a - 9cos(a)) = 0
4e^1.57 - 4e^a + 9cos(a) = 0
8. The final step is to solve the equation for the value of a:
Now you have two equations:
-4e^a + 9cos(a) = 8
4e^1.57 - 4e^a + 9cos(a) = 0
Solve these two equations simultaneously to determine the value of a.
Once you find the value of a, you can substitute it back into the equation F(x) + C to find the full function f(x).
Note: The calculation steps mentioned above can be tedious. You might find it helpful to use an online symbolic mathematics tool like Wolfram Alpha to help with the integration and solving of the equations.
too bad you didn't bother to show your work.
f" = 4e^x - 9sinx
f' = 4e^x + 9cosx + c1
f = 4e^x + 9sinx + c1*x + c2
using the two values of f(x), we have
4+0+0+c2 = 3 ==> c2 = -1
4e^(π/2) + 9 + c1 * π/2 -1 = 0
c1 = -(8+4e^(π/2))/(π/2) = -8/π (2+e^(π/2))
f(x) = 4e^x + 9sinx - 8/π (2+e^(π/2)) x - 1