Derivatives

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1. How can I evaluate the numerical derivative of f(X) at the point x=a using the calculator?

Answer #1: We assume that the function f(X) is stored in the function variable Y₁. First, graph the function in such a way that the number x=a is in the graph window in the x direction. Next, Press[2nd] [CALC] [6] . This will select the derivative option dy/dx and return to the graph window with a blinking cursor on the graph of the function. Move the cursor to the point on the graph which has the value of x equal to the number "a" (or, type the value for "a"). This will set the value of x at the bottom of the screen. Next, pres [ENTER]. The numerical derivative will appear at the bottom of the screen, and the blinking cursor will be on the graph at the point where the derivative was evaluated. The value of the derivative will also be left in the variable "Ans".

Answer #2: On the home screen, enter the instruction nDeriv(Y₁,X,value), where "value" is the value of "a". The numerical value of the derivative will appear on the home screen. The function "nDeriv(" can be found in the [MATH] menu.

2. I know that y=abs(x) is not differentiable at zero, but when I try to find the numerical derivative at 0, instead of the calculator telling me it doesn't exist, it gives me zero. Why?

Answer: When you ask the calculator to do a numerical derivative, it assumes there is one and tries to compute one. The routine that the calculator uses when it does the numerical derivative is: it computes (f(x+h)-f(x-h))/2h. For small values of h this is a good estimate of the derivative. If you apply this to the function f(x)=abs(x) using x=0 to compute the derivative at 0, you get (abs(h)-abs(-h))/2h which is zero. The thing to remember is that the machine always assumes there is a derivative if you ask it to compute one. It cannot tell you if a function is not differentiable.

3. When I try to compute the numerical derivative of y=1/x at zero, I am getting a finite number? How could that be?

Answer: See the answer to the previous question.

4. I have a formula for the function f(x). I used the derivative laws to get a formula for f '(x). How can I check my formula for f '(x) using a TI83 calculator?

Answer: There are many answers to this question. Here is one possible answer:
Put your formula for f(x) into the Y₃ function variable. Put your formula for the derivative f '(x) into the Y₂ function variable. In Y₁, enter the expression nDeriv(Y₃,X,X). (nDeriv can be found in the [MATH] menu. Y₃ is found in the [VARS] menu.) Next, go to [TBLSET] and select "Indpnt:Ask". Finally, press [TABLE] and enter random values for X.

In your table, the columns for Y₁ and Y₂ should, in general, be the same to about 5 or 6 significant figures. If they are not the same, then your formula for f '(x) is probably not correct. (This is because your numerical derivative using f(x) is not giving the same numbers as your function values using f '(x).)

5. I have an equation in the form f(X,Z) = 0. I used implicit differentiation to get a formula for dZ/dX. (The formula for dZ/dX is an expression in Z and X.) How can I check my formula for dZ/dX using a TI83 calculator?

Answer: There are many answers to this question. Here is one possible answer:
Put your formula for f(X,Z) into the Y₃ function variable (as an expression in X and Z). Put your formula for the derivative dZ/dX into the Y₂ function variable (also as an expression in X and Z). In Y₁, enter the expression ( -nDeriv(Y₃,X,X) / nDeriv(Y₃,Z,Z) )/ Y₂. (nDeriv can be found in the [MATH] menu. Y₃ is found in the [VARS] menu.) Remember, the leading minus sign is a small minus. Next, go to the home screen and put a random number into the Z variable. Next, go to [TBLSET] and select "Indpnt:Ask". Finally, press [TABLE] and enter random values for X. (Your values for (X,Z) should be in the domain of f(X,Z).)

In your table, all the numbers in the column for Y₁ should, in general, be equal to the number 1 to about 5 or 6 significant figures. If they are not, then your formula for dZ/dX is probably not correct. (This is because your numerical "implicit derivative" using f(X,Z) (and computed in the function variable Y₁) is not giving the same numbers as your function values using the expression for dZ/dX (which is computed in the function variable Y₂).)

(Note: If your original equation is given in the form f(X,Y) = 0, you should change all the Y variables into Z variables before using the above technique. This is because the TI83 does not allow the use of the variable Y in an expression for a function, for the purpose of generating a table. See question #6 in the section entitled "Curve Sketching".)




Related questions




Q1. How do I enter the variable Y₁ onto the home screen?




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