How do you find an equation for the function #f'(x)=sin4x# whose graph passes through the point (pi/4,-3/4)?

Answer 1

#f(x) = -1/4cos(4x) - 1#.

Use substitution to integrate. Let #u = 4x#. Then #du = 4dx# and #dx= (du)/4#.

Integrate both sides.

#intf'(x) = 1/4intsinu du#
#f(x) = -1/4cosu + C -> "because" intsinxdx = -cosx#
#f(x) = -1/4cos(4x) + C#
Now find the value of #C#. When #x= pi/4#, #y = -3/4#, therefore:
#-3/4 = -1/4cos(4(pi/4)) + C#
#-3/4 = -1/4cos(pi) +C#
#-3/4 = 1/4 +C#
#-3/4 - 1/4 = C#
#C = -1#
Therefore, #f(x) = -1/4cos(4x) - 1#.

Hopefully this helps!

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Answer 2

To find an equation for the function f'(x) = sin(4x) that passes through the point (π/4, -3/4), we can integrate the given function to find the original function f(x).

Integrating f'(x) = sin(4x) with respect to x, we get: f(x) = -1/4 * cos(4x) + C

To determine the value of C, we substitute the given point (π/4, -3/4) into the equation: -3/4 = -1/4 * cos(4(π/4)) + C

Simplifying the equation: -3/4 = -1/4 * cos(π) + C -3/4 = -1/4 * (-1) + C -3/4 = 1/4 + C C = -1

Therefore, the equation for the function f(x) = -1/4 * cos(4x) - 1, which is the antiderivative of f'(x) = sin(4x), and it passes through the point (π/4, -3/4).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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