Nderiv (Expression,Variable,Value) K 1 Nderivative(Expression,X=Value)

Calculator Lab 4 Rates of Change – The Derivative

Name ______Date ______

Calculator Functions Ti-Nspire nDeriv(expression,variable,value) k 1 nDerivative(expression,x=value) CALC MENU 4: Calculus 1: Numerical Derivative at a Point

Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. One of these problems involved tangent lines to curves or the rate at which one quantity changes with respect to another.

1. Consider the function f(x) = 2x + 5. a) analytically compute the slope ______y b) compute the average rate of change ( ) between x = -5 and x = -1 x ______

c) the average rate of change of y with respect to x can also be found using the f (x  x)  f (x) following formula F(x) = where x = -5 and x = 4. Using it x what results do you obtain? ______

d) GRAPH g(x) = x2 + 8x +10 and f(x) = 2x + 5 on the same screen. What type of line is f(x) to the curve g(x)? ______

e) What relationship is there between a secant line to a curve and the average rate of change over the same interval? ______

______

f) If we would continue to draw secant lines whose points of intersection with the curve became closer and closer together, what would happen to  x?

______

Allowing x to become increasing small or allowing x to approach 0 would provide a way for us to compute the instantaneous rate of change at a point on the curve and the corresponding line at that point would be tangent to the curve. 2. Using the formula from question 1 step c and the curve f(x) = x2 + 8x + 10, complete the following table to investigate the instantaneous rate of change when x = -1.

Inte f (1  x)  f (1) Average rate of change = rval Size ( x ) x 0.1

0.0001 a) What happens as x gets smaller and smaller?______b) What would you guess the instantaneous rate of change at x = -1 to be? ______c) What would you guess the slope of the curve at x = -1 to be? ______d) What would you guess the slope of the line tangent to the curve at x = -1 to be? ______e) Using the value that you found for the slope and (-1, f(-1)) as your point, determine the equation for the line through that point with that slope.

______f) GRAPH the equation you found in e) and the equation for the curve f(x) on the same screen. Show your results below (be sure to label). 3. Using the formula from question 1 step c and the curve f(x) = 4x3 – 3x -1, complete the following table for the instantaneous rate of change when x = 2.

Inte f (2  x)  f (2) Average rate of change = rval Size ( x ) x 0.1

0.0001 a) What happens as x gets smaller and smaller?______b) What would you guess the instantaneous rate of change at x = 2 to be? ______c) What would you guess the slope of the curve at x = 2 to be? ______d) What would you guess the slope of the line tangent to the curve at x = 2 to be? ______e) Using the value that you found for the slope and (2, f(2)) as your point, determine the equation for the line through that point with that slope.

______f) GRAPH the equation you found in e) and the equation for the curve f(x) on the same screen. Show your results below (be sure to label). 4. Instead of using the table of values to determine the slope of the tangent line, we can use the limit process. The resulting equation is

f (c  x)  f (c) lim = m where m is the slope of the curve at point (c , f(c) ) and x0 x the line tangent to the curve at x = c. This equation is known as the derivative of f and is also denoted by f ‘ (x).

Using this approach with the equation f(x) = 2x2 at the point (1 , 2), we would have f (1 x)  f (1) lim x0 x 2(1 x)2  2(1)2 lim x0 x 2  4x  2x2  2 lim x0 x 4x  2x2 lim x0 x

lim 4 + 2 x = 4 x0 Apply this process to the equation from question 2 (f(x) = x2 + 8x + 10, x = -1). What are your results?

______

5. The TI-83 has 2 built in functions for calculating the derivative at a point. Perform the following and record your findings.

a) Enter Y= x2 + 8x + 10 2nd CALC 6 –1 ENTER ______

b) from the home screen enter MATH 8 x2 + 8x + 10, x, -1) ENTER ______

c) Verify your finding using the analytical approach from question 4.

For the Ti-Nspire use MENU 4: Calculus 1: Numerical Derivative at a Point o 6. Complete the table for the following functions using what you have learned in the f (x  x)  f (x) question above. F(x) = x

Function X Derivative Derivative Limit F(x) x Limit F(x) x = Using CALC Using nDeriv (x value)-  (x value)+ or Calculus Menu a) F(x) = e 3x 0 b) F(x) = sin x 1 c) F(x) = 4x3 + 2x – 4 1 d) F(x) = x2 – 3x – 7 -1 e) F(x) = x2 – 3x – 7 2 f) F(x) = | x – 2 | 2

1 g) F(x) = x 3 0 h) F(x) = [[x]] 1

Now for each of the functions above, do the following and record your results. A tangent line to the graph at the point specified will be drawn. Record the equation for the tangent line under the box. (Don’t forget the scale!)

Ti-83/84 Ti-Nspire a) enter function in Y= and GRAPH a) on Graph page enter function in f1 b) 2nd DRAW 5 (x= value from table) b) on Calculator page enter f1(value) ENTER c) on Graph enter value in f2 d) find intersection of f1and f2 at value e) select f1 graph f) Menu 8: 1: 7: at point of intersection g) Menu 1: 8: with hand on tangent line

a) b) c) d)

g) h) Looking at the information in the table and the graphs above, under what conditions does the Ti-83/84 CALC and nDeriv commands give invalid results or under what conditions does the Ti-Nspire fail to give you a numerical answer?

1) ______

2) ______

3) ______

Technology Stretch

Let f be the function given by f(x) = 3 cos x. As shown, the graph crosses the y-axis at point P and the x-axis at point Q. a) Write an equation for the line passing through points P and Q.

______b) Write an equation for the line tangent to the graph of f at point Q. Show the analysis that leads to your equation.

______c) Find the x-coordinate of the point on the graph of f, between points P and Q, at which the line tangent to the graph of f is parallel to line PQ. ______