Suppose that u and v are differentiable functions

butleraliyah21

butleraliyah21

Answered question

2022-08-03

Suppose that u and v are differentiable functions of x and in particular that š‘¢š‘¢(0) = 5, š‘¢š‘¢ā€² (0) = āˆ’3, š‘£š‘£(0) = āˆ’1, š‘£š‘£ā€² (0) = 2 

Find the values of the following derivatives at x=0 š‘‘š‘‘ š‘‘š‘‘š‘‘š‘‘ (š‘¢š‘¢š‘£š‘£), š‘‘š‘‘ š‘‘š‘‘š‘‘š‘‘ (7š‘£š‘£ āˆ’ 2š‘¢š‘¢), š‘‘š‘‘ š‘‘š‘‘š‘‘š‘‘ ļæ½ š‘¢š‘¢ š‘£š‘£

Answer & Explanation

nick1337

nick1337

Expert2023-05-29Added 777 answers

To solve the given problem, let's start by using the notation uā€²(x) to represent the derivative of function u(x) with respect to x. Similarly, vā€²(x) represents the derivative of function v(x) with respect to x.
Given information:
u(0)=5
uā€²(0)=āˆ’3
v(0)=āˆ’1
vā€²(0)=2
Now, we need to find expressions for u(x) and v(x) based on the given information. We'll use the fact that uā€²(x) and vā€²(x) are the derivatives of u(x) and v(x), respectively.
Let's integrate uā€²(x) to find u(x). We have:
uā€²(x)=āˆ’3
Integrating both sides with respect to x, we get:
āˆ«uā€²(x)dx=āˆ«āˆ’3dx
Integrating the left side gives us:
u(x)=āˆ’3x+C1
where C1 is the constant of integration.
Since u(0)=5, we can substitute this into the equation:
5=āˆ’3(0)+C1
Simplifying, we find:
C1=5
Therefore, the expression for u(x) is:
u(x)=āˆ’3x+5
Next, let's integrate vā€²(x) to find v(x). We have:
vā€²(x)=2
Integrating both sides with respect to x, we get:
āˆ«vā€²(x)dx=āˆ«2dx
Integrating the left side gives us:
v(x)=2x+C2
where C2 is the constant of integration.
Since v(0)=āˆ’1, we can substitute this into the equation:
āˆ’1=2(0)+C2
Simplifying, we find:
C2=āˆ’1
Therefore, the expression for v(x) is:
v(x)=2xāˆ’1

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