The objective of this lab is to introduce students to the concept of antiderivatives and their applications in calculus. By working through a series of exercises, students will develop an understanding of how to find antiderivatives, verify their results using computational tools, and apply these concepts to real-world scenarios, such as calculating areas under curves and solving kinematics problems.
By completing this lab, students will gain practical experience in antidifferentiation, understand its applications in different contexts, and appreciate the interplay between derivatives and integrals.
The goal of differentiation is to find the derivative \(f'\) of a given function \(f\). The reverse process called antidifferentiation, is equally important: Given a function \(f\), we look for an antiderivative function \(F\) whose derivative is \(f\); that is, a Function \(F\) such that \(F'=f\)
Antiderivative
A function F is an antiderivative of \(f\) on an interval \(I\) provided \(F'(x) = f(x)\), for all \(x\) in \(I\)
Without using maxima, but using the definition, calculate the antiderivative of the following functions:
Using the function integrate(f(x),x) verify each of your previous results.
For the following functions calculate \(A(x)\):
For the following functions, by using the function integrate(f(x),x,begin,end), calculate the area under the curve in the interval.
Using derivatives, starting from the displacement equation, for a 1D projectile, deduce the equations of velocity and Acceleration as function of time.
\[s(t) = x_0 + v_0t - g\frac{t^2}{2}\]
Solution:
(%i1) define(s(t),x0 + v0*t - g*t^2/2);\[\mathtt{(\textit{%o}_{1})}\quad s\left(t\right):=x_{0}+t\,v_{0}-\frac{g\,t^2}{2}\]
(%i2) define(v(t),diff(s(t),t,1));\[\mathtt{(\textit{%o}_{2})}\quad v\left(t\right):=v_{0}-g\,t\]
(%i3) define(a(t),diff(v(t),t,1));\[\mathtt{(\textit{%o}_{3})}\quad a\left(t\right):=-g\]
(%i4) diff(s(t),t,2);\[\mathtt{(\textit{%o}_{4})}\quad -g\]
Using anti-derivatives, starting from the acceleration equation, for a 1D projectile, deduce the equations of velocity and displacement as function of time.
(%i5) define(a(t),-g);\[\mathtt{(\textit{%o}_{5})}\quad a\left(t\right):=-g\]
(%i6) define(v(t),integrate(a(t),t)+v0);\[\mathtt{(\textit{%o}_{6})}\quad v\left(t\right):=v_{0}-g\,t\]
(%i7) define(s(t),integrate(v(t),t)+x0);\[\mathtt{(\textit{%o}_{7})}\quad s\left(t\right):=x_{0}+t\,v_{0}-\frac{g\,t^2}{2}\]
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