Find and classify the relative maxima and minima of this function
f(x)= definite integral sign where a=0 and b=x.
t^2-4/(1+(cos^2(t)) dt
what should my value for u be? is it just t. not sure how to even tackle this problem.
The clue here is that we want to find extrema of f(x), which involves finding the derivative of f. Since f(x) is defined as an integral, we don't really have to do the integration. We just apply the rules for differentiating under the integral sign. (See wikipedia, and scroll down for some examples)
f(x) = ∫[0,x] (t^2-4)/(1+(cos^2(t)) dt
so,
df/dx = (x^2-4)/(1+cos^2 x)
So, f'=0 when x^2-4 = 0, since the bottom is never zero.
Obviously the extrema are at x=2,-2.
Thank you!
To find and classify the relative maxima and minima of the given function f(x) = ∫[a=0 to b=x] (t^2 - 4)/(1 + cos^2(t)) dt, you need to find the derivative of the function and then solve for critical points.
To find the derivative of the integral, we will use the Leibniz rule for differentiating under the integral sign. The derivative of the function with respect to x is given by:
f'(x) = d/dx [∫[a=0 to b=x] (t^2 - 4)/(1 + cos^2(t)) dt]
Let's find this derivative step-by-step:
Step 1: Set up the notation using a function g(t, x) as follows:
g(t, x) = (t^2 - 4)/(1 + cos^2(t))
Step 2: Differentiate g(t, x) with respect to x:
∂g/∂x = 0 (since there is no x term in the integrand)
Step 3: Integrate the above partial derivative with respect to t:
∫[a=0 to b=x] (∂g/∂x) dt = ∫[a=0 to b=x] 0 dt
This integration amounts to a constant, so the result of the integration is equal to C.
Therefore, f'(x) = C.
Now, to solve for critical points, we need to set the derivative equal to zero and solve for x:
C = 0
Since C is a constant, it has no effect on x. Hence, there are no critical points or relative maxima/minima for the given function f(x).
Regarding your question about the value for "u," it appears that you are working with indefinite integration rather than definite integration in this case. Since the function is already given as a definite integral from a=0 to b=x, there is no need to introduce a variable "u" as we typically do in indefinite integration using the u-substitution method.
Therefore, in this particular problem, you do not need to consider a value for "u".
To find and classify the relative maxima and minima of the given function, we first need to find its derivative. Let's start by finding the derivative of the function and then proceed step by step:
1. Differentiate the function using the Fundamental Theorem of Calculus:
Differentiating the integral function with respect to x will give us the derivative of the function.
f'(x) = d/dx ( ∫[0 to x] (t^2 - 4 / (1 + cos^2(t))) dt )
2. To differentiate the integral function, we need to apply the Leibniz rule for differentiating under the integral sign:
Let F(x,t) = t^2 - 4 / (1 + cos^2(t)).
Then the derivative of the integral function can be expressed as follows:
f'(x) = d/dx ( ∫[0 to x] F(x,t) dt )
= F(x,x) + ∫[0 to x] ∂F(x,t) / ∂x dt,
where F(x,x) is the value of F(x,t) when t is replaced by x, and ∂F(x,t) / ∂x is the partial derivative of F(x,t) with respect to x.
3. Find F(x,x):
Substitute x for t in the expression for F(x,t):
F(x,x) = x^2 - 4 / (1 + cos^2(x)).
4. Find ∂F(x,t) / ∂x:
Differentiate F(x,t) with respect to x:
∂F(x,t) / ∂x = 2x.
5. Integrate ∂F(x,t) / ∂x with respect to t:
Integrate 2x with respect to t to obtain:
∫[0 to x] ∂F(x,t) / ∂x dt = 2x * (∫[0 to x] dt)
= 2x * x
= 2x^2.
6. Substitute the calculated values back into the expression for f'(x):
f'(x) = F(x,x) + ∫[0 to x] ∂F(x,t) / ∂x dt
= (x^2 - 4 / (1 + cos^2(x))) + 2x^2
= x^2 + 2x^2 - 4 / (1 + cos^2(x)).
We have now found the derivative of the function f(x) with respect to x. To classify the relative maxima and minima, we need to find the critical points, where the derivative is zero or undefined. Next, we can analyze the sign of the derivative to determine the increasing and decreasing intervals, which will help identify the relative maxima and minima.