Derivatives_and_Integrals.sagews

Defining a function in SageMath is quite easy. Give the function a name, list the arguments in parentheses, then type out the function.

Defining a function

• Function name must start with a letter and can contain numbers and underscores (no spaces)

• List the arguments of the function. These do not need to be defined as variables

  • For example y(t)=y_0+v_0*t+1/2*a*t^2

• Functions can be evaluated by entering values for each argument

  • For example: y(1) yields the value of y when t=1.

x(x_0,v_0x,t)=x_0+v_0x*t  #Position equation

x(0,2,8)  #Evaluate the position when x_0=0, v_0x=2, and t=8

x(10,-1,4)  #Evaluate the position when x_0=10, v_0x=-1, and t=4

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Taking Derivatives

• Can either use diff(f(x),x) or f(x).diff(x)

• diff() or derivative() are identical

• Specify variable to differentiate with respect to

• Indicate the order of the derivative

y_0,v_0y,a_y=var('y_0,v_0y,a_y')   #Define all variables

y(t)=y_0+v_0y*t+1/2*a_y*t^2  #Define y(t)

### A couple of different ways to take the derivative of y with respect to t
y(t).diff(t)
diff(y(t),t)
derivative(y(t),t)

### Second derivative of y with respect to t should return the acceleration
y(t).diff(t,2)
diff(y(t),t,2)

a_y*t + v_0y a_y*t + v_0y a_y*t + v_0y a_y a_y

The SageMath derivative returns a function so you can assign it a name or evaluate it at a point just like with other functions.

diff(y(t),t)(t=1)

v(t)=diff(y(t),t)
v(1)

a_y + v_0y a_y + v_0y

Plotting Functions

• Use the plot() command to create a plot

  • Arguments of plot() include the function to plot, the variable to plot against, and the variable range

    • plot(y(t),t,0,10) plots the function y(t) vs. t from t=0 to t=10

• Plots can be given variable names which is useful for modifying features of the plot

  • For example: P1=plot(y(t),t,0,10)

• The show() command displays the plot on the screen

  • For example: P1.show()

y_plot(t)=y(t).substitute(a_y=-9.8,v_0y=20,y_0=10)  #Must substitute numerical values for all variables (except t) in order to plot

pos_plot=plot(y_plot(t),t,0,10,title='Position vs. Time')
pos_plot.show()  #Display pos_plot on the screen

Try It Yourself

• Plot the function $F=G \frac{m_1 m_2}{r^2}$ vs. $r$

  • Use .substitute() to input values for G, $m_1$, and $m_2$

  • Choose a minimum value of $r$ so graph does not blow up near the origin

#Input your work here

Plotting Multiple Functions

• Plots can be 'added' together for display on the same graph