Calculus

# Differential Equations - dy/dx = f(x)

If function $$f(x)$$ satisfies $\frac{df(x)}{dx}=(3x+4)(2-x) \text{ and } f(-1)=18,$ what is $$f(x)?$$

If $$f'(x)=9x^2$$ and $$f(2)=31,$$ what is $$f(x)?$$

If $f'(x)=\sin x-2\cos x, f(\pi)=9,$ what is $$f(x)?$$

Suppose that the acceleration $$a(t)$$ of a particle at time $$t,$$ is given by the formula $$a(t)=4t-3.$$ If $v(1)=8, f(2)=5,$ where $$v(t)$$ and $$f(t)$$ are the velocity and position function of the particle, respectively, what is the position of the particle when $$t=1 ?$$

The velocity at time $$t$$ of a particle moving along the $$x$$-axis is given by the formula $v(t)=8t^3-9t^2.$ If the initial position of the particle on the $$x$$-axis is $$x=9$$, what is its position when $$t=2 ?$$

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