Derivatives:

Using the TI-83 or higher, approximate f'(0.75) for f(x)= 0.5x³ + 2x + 0.69
Then calculate the actual value of f'(0.75) and compare this with the approximation.

First graph f(x)= 0.5x³ + 2x + 0.69

1. Press Y= key.
2. Type in the equation with the following keys: . 5 X ^ 3 + 2 X + . 6 9
3. Press the ZOOM key and hit 6 for zoom standard. The graph should appear and look something like this:

Approximate f'(0.75) on the calculator

1. Hit 2nd MODE to go back to the home screen.
2. Press MATH and 8 so you have the nDeriv( on your home screen.
3. Press the VARS key, scroll right to Y-VARS, press 1 and press 1 again to select the Y1 function we entered.
4. Now enter , X , 0 . 7 5 , 0 . 0 0 0 1 ) and hit ENTER.
5. The calculator has approximated the value of the derivative of f(x) at the point x = 0.75 to be ~2.843750005
   

Find the derivative of f(x)

1. Find f'(x) by using the Chain Rule: f'(x) = 1.5x² + 2
2. Graph this function along with the original by pressing Y= again, and entering 1.5 X ^ 2 + 2 next to Y2=.
3. Then hit the GRAPH key. The graph should look something like this:

Calculate the exact value of f'(0.75)

1. Plug 0.75 into the derivative equation: f'(0.75) = 1.5 (0.75)² + 2 = 2.84375
2. Compare this with the approximation: Pretty darn close!!!

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Integrals:

Using the TI-83 or higher, find the area between the curves f(x)= x³-3x +1 and g(x)= 2sin(x).

First graph f(x) and g(x)

1. Press Y= key.
2. Type in f(x) next to Y1= with the following keys: X ^ 3 - 3 X + 1
3. Type in g(x) next to Y2= with the following keys: 2 SIN X )
   
4. Press the ZOOM key and hit 6 for zoom standard. The graph should appear and look something like this:

Find the solutions to the system of equations; in other words, find where the functions intersect

1. Press 2nd CALC (TRACE key).
2. It will ask for the first curve, so move the cursor close to the first intersection point. Press ENTER.
3. On the other curve move the cursor close to the intersection, and press ENTER again.
4. To guess the intersection point, press ENTER one more time. The coordinates of the point are displayed on the screen.
   
5. Repeat steps 1-3 to determine the other two points of intersection.

x1 = -2.0788
x2 = 0.2022
x3 = 1.8678

Find the area of the two regions between the two curves

1. Hit 2nd MODE to go back to the home screen.
2. For the left region, subtract the integral of g(x) from f(x) from x1 to x2.
3. Press MATH 9 and fnInt( will appear. Press the VARS key, scroll right to Y-VARS, press 1 and press 1 again to select the Y1 function we entered.
4. Now enter , X , -2.0788 , 0.2022 ) followed by the minus sign -
5. Repeat step 3 but enter the Y2 function instead of Y1 (press Y-VARS, 1, 2). Then enter , X , -2.0788 , 0.2022 ) and ENTER.
6. This is the area for the left-most region is ~ 6.9657
7. For the right region, subtract the integral of f(x) from g(x) from x2 to x3.
   
8. Repeat the same procedure above to find this area. This time use fnInt(y2, x, x2, x3) - fnInt(y1, x, x2, x3).
9. The area for the right-most region is ~3.0084
   

Add the two regions together to get the total area between the curves f(x) and g(x): 9.9740






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Michelle M. Beckman
beckman.michelle@gmail.com
Updated December 11, 2008.