Aim:
\[1.\ \text{Graph the polynomial function}\ f(x)\ =\ x^2\ -\ 5x\ +\ 6.\ \hspace{15cm}\\ \text{Find the limit of f(x) at x = 1}\ \hspace{10cm}\]
\[2.\ \text{Graph the rational function}\ f(x)\ =\ \frac{x^2\ -\ 5x\ +\ 6}{x+2}.\ \hspace{15cm}\\ \text{Find the limit of f(x) at x = 4}\ \hspace{10cm}\]
\[3.\ \text{Graph the functions}\ \frac{sin\ 3x}{2x}\ and\ \frac{tan\ 3x}{2x}.\ \hspace{15cm}\\ \text{Evaluate}\ \lim\ _{x\ \to\ 0}\ \frac{sin 3x}{2x}\ and\ \lim\ _{x\ \to\ 0}\ \frac{tan 3x}{2x}\ \hspace{10cm}\]
\[4.\ \text{Graph the functions}\ x^3, \text{sin x, cos x, tan x,}\ \hspace{15cm}\\ \text{cosec x, sec x, cot x,}\ e^x\ log x\ and\ 5.\ \hspace{10cm}\\ \text{Find their first derivative and second derivative}\ \hspace{15cm}\]
Procedure:
\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{18cm}\]
\[\color{green}{Step\ 2:}\ \text{To evaluate the limit of the polynomial functions.}\ \hspace{18cm}\\ \text{Menu bar → Options → Rounding → 4 decimal places}\ \hspace{10cm}\]
\[\color{green}{Step\ 3:}\ \text{To draw the graph of the function}\ x^2\ -\ 5x\ +\ 6\ \hspace{18cm}\\ \text{by using the input bar type x^2-5x+ 6}\ \text{and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 4:}\ \text{Create a slider a with minimum = -5 and maximum = 5}\ \hspace{18cm}\]
\[\color{green}{Step\ 5:}\ \text{Plot the point A on the function graph}\ \hspace{18cm}\\ \text{by using the input bar type A=(a,f(a)) and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 6:}\ \text{Create a slider h with minimum = 0 and maximum = 1}\ \hspace{18cm}\]
\[\color{green}{Step\ 7:}\ \text{To plot the point L(Left) on the curve f(x)}\ \hspace{18cm}\\ \text{by using the input bar type L=(a-h,f(a-h)). and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 8:}\ \text{To plot the point R(Right) on the curve f(x)}\ \hspace{18cm}\\ \text{by using the input bar type R=(a+h,f(a+h)). and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 9:}\ \text{To Calculate the value of the function at a point,}\ \hspace{18cm}\\ \text{by using the input bar to type LHV=f(a-h). Similarly, RHV=f(a+h)}\ \hspace{14cm}\]
\[\color{green}{Step\ 10:}\ \text{Create an input box with label h = and link with the slider h}\ \hspace{16cm}\]
\[\color{green}{Step\ 11:}\ \text{Now we assign values for h as}\ \hspace{18cm}\\ \text{1,0.5,0.2,0.1,0.08,0.04,0.03,0.02,0.01,0}\ \hspace{14cm}\\ \text{and observe the values of a-h, a+h, LHV and RHV}\ \hspace{14cm}\]
\[\color{green}{Step\ 12:}\ \text{Create an input box for function and link with f(x).\ and also Create an input box for a and link with a=1}\ \hspace{10cm}\]
To evaluate the limit of the rational functions.
\[\color{green}{Step\ 13:}\ \text{Change the function}\ \frac{x^2\ -5x+6}{x+2}\ \hspace{18cm}\\ \text{in the function input box and}\ \hspace{14cm}\\ \text{Enter 4 in the input box of a.}\ \hspace{10cm}\]
\[\color{green}{Step\ 14:}\ \text{Now we assign values for h as}\ \hspace{18cm}\\ \text{1,0.5,0.2,0.1,0.08,0.04,0.03,0.02,0.01,0}\ \hspace{14cm}\\ \text{and observe the values of a-h, a+h, LHV and RHV}\ \hspace{14cm}\]
\[\text{To evaluate the limits}\ \lim\ _{x\ \to\ 0}\ \frac{sin 3x}{2x}\ and\ \lim\ _{x\ \to\ 0}\ \frac{tan 3x}{2x}\]
\[\color{green}{Step\ 15:}\ \text{Change the function}\ \frac{sin 3x}{2x}\ \hspace{18cm}\\ \text{in the function input box and}\ \hspace{14cm}\\ \text{Enter 0 in the input box of a.}\ \hspace{10cm}\]
\[\color{green}{Step\ 16:}\ \text{Right click on the graphics view}\ \rightarrow \hspace{18cm}\\ Graphics\ \rightarrow\ Preferences\ \rightarrow\ \hspace{14cm}\\ Graphics\ \rightarrow\ xAxis\ \rightarrow\ Distance\ \rightarrow\ \frac{\pi}{2}\ \hspace{14cm}\]
\[\color{green}{Step\ 17:}\ \text{Now we assign values for h as}\ \hspace{18cm}\\ \text{1,0.5,0.2,0.1,0.08,0.04,0.03,0.02,0.01,0}\ \hspace{14cm}\\ \text{and observe the values of a-h, a+h, LHV and RHV}\ \hspace{14cm}\]
\[\color{green}{Step\ 18:}\ \text{Change the function}\ \frac{tan 3x}{2x}\ \hspace{18cm}\\ \text{in the function input box and}\ \hspace{14cm}\\ \text{Enter 0 in the input box of a.}\ \hspace{10cm}\]
\[\color{green}{Step\ 19:}\ \text{Now we assign values for h as}\ \hspace{18cm}\\ \text{1,0.5,0.2,0.1,0.08,0.04,0.03,0.02,0.01,0}\ \hspace{14cm}\\ \text{and observe the values of a-h, a+h, LHV and RHV}\ \hspace{14cm}\]
To evaluate the first and second derivatives of the functions.
\[\color{green}{Step\ 20:}\ \text{Open new Geogebra classic 5 window}\ \hspace{18cm}\]
\[\color{green}{Step\ 21:}\ \text{To draw the graph of the function}\ f(x)=sin x\ \hspace{18cm}\\ \text{by using the input bar type f(x)=sin x}\ \text{and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 22:}\ \text{find the first derivative of the given function f(x)}\ \hspace{18cm}\\ \text{by using the input command f′(x).}\ \hspace{14cm}\\ \text{and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 23:}\ \text{find the second derivative of the given function f(x)}\ \hspace{18cm}\\ \text{by using the input command f”(x).}\ \hspace{14cm}\\ \text{and press the Enter key}\ \hspace{14cm}\]
\[\color{green}{Step\ 24:}\ \text{Create an input box for function and link with f(x).}\ \hspace{17cm}\]
\[\color{green}{Step\ 25:}\ \text{using the input box Enter the functions}\ \hspace{18cm}\\ x^3,\ \text{sinx, cosx, tax, cosecx, secx,cotx,}\ e^x\ and\ logx\ \hspace{14cm}\\ \text{and record the expressions of first derivative}\ \hspace{14cm}\\ \text{and second derivatives of the given functions.}\ \hspace{14cm}\]
Output:
Limit of the polynomial functions
| h | a-h | a+h | LHV | RHV |
| 1 | 0 | 2 | 6 | 0 |
| 0.5 | 0.5 | 1.5 | 3.75 | 0.75 |
| 0.2 | 0.8 | 1.2 | 2.64 | 1.44 |
| 0.1 | 0.9 | 1.1 | 2.31 | 1.71 |
| 0.08 | 0.92 | 1.08 | 2.0241 | 1.9761 |
| 0.04 | 0.96 | 1.04 | 2.1216 | 1.8816 |
| 0.03 | 0.97 | 1.03 | 2.0909 | 1.9109 |
| 0.02 | 0.98 | 1.02 | 2.0604 | 1.9404 |
| 0.01 | 0.99 | 1.01 | 2.0301 | 1.9701 |
| 0 | 1 | 1 | 2 | 2 |
| 2 |
Limits of the rational functions
| h | a-h | a+h | LHV | RHV |
| 1 | 3 | 5 | 0 | 0.8571 |
| 0.5 | 3.5 | 4.5 | 0.1364 | 0.5769 |
| 0.2 | 3.8 | 4.2 | 0.2483 | 0.4258 |
| 0.1 | 3.9 | 4.1 | 0.2898 | 0.3787 |
| 0.08 | 3.992 | 4.008 | 0.3298 | 0.3369 |
| 0.04 | 3.96 | 4.04 | 0.3157 | 0.3513 |
| 0.03 | 3.97 | 4.03 | 0.3201 | 0.3467 |
| 0.02 | 3.98 | 4.02 | 0.3245 | 0.3423 |
| 0.01 | 3.99 | 4.01 | 0.3289 | 0.3378 |
| 0 | 4 | 4 | 0.3333 | 0.3333 |
| 0.3333 |
Limits of the trigonometric functions
| h | a-h | a+h | LHV for | RHV | LHV for | RHV |
| 1 | -1 | 1 | 0.0706 | 0.0706 | -0.0713 | -0.0713 |
| 0.5 | -0.5 | 0.5 | 0.9975 | 0.9975 | 14.1014 | 14.1014 |
| 0.2 | -0.2 | 0.2 | 1.4116 | 1.4116 | 1.7103 | 1.7103 |
| 0.1 | -0.1 | 0.1 | 1.4776 | 1.4776 | 1.5467 | 1.5467 |
| 0.08 | -0.08 | 0.08 | 1.4856 | 1.4856 | 1.5295 | 1.5295 |
| 0.04 | -0.04 | 0.04 | 1.4964 | 1.4964 | 1.5072 | 1.5072 |
| 0.03 | -0.03 | 0.03 | 1.498 | 1.498 | 1.5041 | 1.5041 |
| 0.02 | -0.02 | 0.02 | 1.4991 | 1.4991 | 1.5018 | 1.5018 |
| 0.01 | -0.01 | 0.01 | 1.4998 | 1.4998 | 1.5005 | 1.5005 |
| 0 | 1.5 | 1.5 | 1.5 | 1.5 | ||
| 1.5 | 1.5 |
first derivative and second derivative
| Function f(x) | First derivative f′(x) | Second derivative f”(x) |
| 6x | ||
| sin x | cos x | – sin x |
| cos x | – sin x | – cos x |
| tan x | 2 tan x + 2 | |
| cosec x | ||
| log x | ||
| 5 | 0 | 0 |
