# Differentiation/Integration [closed]

I am given $$v=6-4x$$ where $$v$$ is velocity and $$x$$ is position of the body. I am now asked to find displacement between $$t=1.4$$, distance covered from $$t=1.4$$, average velocity from $$t=2.5$$, average acceleration, etc. I am not able to find an approach to these kind of problems where $$v$$ is not given in terms of $$t$$, or say, it is given like $$v=kt^2$$ or like $$v=kx$$. How should I start these kind of problems and develop a thinking for these? Please help me.

First thing to do is to write $$v = dx/dt$$. That's by definition. Your formula is now $$dx/dt = 6-4x$$. This is a separable differential equation. Solving this yields an expression for $$x$$ in terms of $$t$$. The rest of the problem can now be solved.
• @AyushBhardwaj once you have $x$ as a function of $t$, you can calculate the average velocity between (say) $t=0$ and $t=1$ by calculating where the particle is at those times, and dividing by the time taken to get there (1s in this case). – Allure Jun 16 at 3:09
• Remember that the average value of some function $f(x)$ on $[a,b]$ is $$\bar f_{[a,b]}=\dfrac1{b-a}\int_a^b f(x) \ \mathrm dx$$ – user256872 Jun 16 at 3:11
• @AyushBhardwaj The average speed is the distance traversed over time; the average of $x=(6-v)/4$ is $(6-\bar{v})/4$. – J.G. Jun 16 at 6:07