AP Calculus BC
Intro to the Derivative
Name:__________________________
Numerical derivatives
The term numerical derivative refers to any numerical approximation of a derivative value at a
single point, or any numerically-constructed approximation of an entire derivative function.
Your calculator has built-in numerical derivative capabilities, as described below. The textbook
uses the name NDER when it refers to numerical derivatives. (page numbers/problems are for
the new edition of the book)
In the following material, calculator instructions are for the TI-83/84. Other models have
equivalent features, but keys and function names may vary a bit, so it might be necessary to
consult a calculator manual.
Numerical derivative at an x-value
Your calculator has two methods for finding the derivative of a function at a single x-value.
One method uses a function called nDeriv; the other method uses the graph.
Example problem. Given f(x) = x2, find a numerical estimate of f "(3) .
Method using nDeriv: The function nDeriv is the 8th choice under the MATH key.
Type nDeriv(X^2,X,3) ENTER, and you should get an answer of 6.
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Method using Graph: Press Y= and enter the function Y1=X^2 . Then, on the CALC menu
(2nd TRACE), select 6:dy/dx. The graph will appear, and now you can enter the desired
x-value. Type 3 and press ENTER. At the bottom of the display you should see dy/dx = 6.
Note that the graph method will only work if the x-value lies within the displayed window.
Sometimes you may need to enlarge the window to include your desired x-value.
Generally, the nDeriv method is easier if all you need is a numerical value; the graph method is
helpful when you already have the graph on the screen.
1. For f(x) = ex, find numerical estimates of f "(1) and f "(2) . Try using both of the methods
described above.
2. For f(x) = ln x, find numerical estimates of f "(2) , f "(5) , and f "(10) . Based on the answers,
and further numerical experimentation
if necessary,
can you guess what function is the
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derivative of f(x) = ln x ?
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Intro to the Derivative
page 2
Finding numerical derivatives without a special calculator feature
Read Derivatives on a Calculator on pp. 113–114 of your textbook (you can stop before
Exploration 2). It explains a standard method for calculating numerical derivatives, using a
symmetric difference quotient, which you should recognize as a variation on the limit definition
of derivative.
3. For f(x) = ex, estimate f "(2) using a symmetric difference quotient. [Follow the method of
Example 2 on page 113.] Compare your answer to the value you found in problem 1.
4. For f(x) = ln x, estimate f "(5) using a symmetric difference quotient.
€ 38 on page 117.
5. Do 3.2 Exercise
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Graphing a numerical derivative function
Your calculator is capable of calculating many numerical derivative values of the same function,
and using these to construct an approximation of the derivative function graph.
Example. Using the calculator, construct an approximate graph of the derivative of f(x) = sin x.
Method: Enter these formulas on the Y= screen.
Y1=sin(X)
Y2=nDeriv(Y1,X,X)
To type Y1, press VARS, YVARS
Now press ENTER. The second graph should look like cos(x).
6. Use nDeriv to graph the derivative of the functions below. If possible, identify the derivative
function by looking at the graph (eg, can you tell what f "(x) is?).
a. y = 0.25x4
c. y =
x⋅ x
2
b. y = –cos(x)
d. €y = –ln |cos(x)|
e. y = 2x cos(2x) + 4x sin(x)
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7. Use your calculator to construct an approximate graph of the derivative of f(x) = ln x.
Does the graph appear to confirm your conjecture from problem 2?
8. Use your calculator to construct an approximate graph of the derivative of f(x) = ex.
What function does this derivative appear to be?
9. Do 3.4 Exercises 47 (page 142) and 29 (page 140; note that marginal just means derivative of
in business applications.)