Centered Difference Quotients Another formula that is used to approximate the derivative of a function at a point is the centered difference quotient (CDQ) f '(a) = 4. a + h a h f( ) f( )_ (2) 2h Again consider f(x) = J; . a. Graph f near the point (4, 2) and let I: : 1/ 2 in the centered difference quotient. Show the line whose slope is computed by the centered difference quotient and explain why the formula approximates f '(4). b. Use the centered difference formula to approximate f '(4) by completing Table 2. I h Approximation Error | 0.1 0.01 0.00] | 0.0001 Table2 c. Explain why it is not necessary to use negative values of h in Table 2. d. Compare the accuracy of the derivative estimates in part (b) with those found in Steps 2 and 3. Use the CDQ (2) and a table similar to Table 2 to nd a good approximation to f '(0) for x) = (1 + x)". Use the CDQ (2) and a table similar to Table 2 to nd a good approximation to f'(rz/6) for f(x) = sin 1:. Table 3 gives the distance t) fallen by a smokejumper 1 seconds after she opens her chute. a. Use the forward difference quotient (1) with h = 0.5 to estimate the velocity of the skydiver at I = 2 s. b. Repeat part (a) using the centered difference uotient 2 . t (seconds) g) (feet) 0 0 0.5 4 1.0 15 1.5 33 2.0 55 2.5 81 3.0 109 3.5 138 4.0 169 Table 3 Computer Rounding Error Using difference approximations to approximate derivatives with a computer or calculator is prone to rounding errors. These errors occur when a calculator rounds a number before using it in an arithmetic calculation. Such rounding may lead to remarkably inaccurate results. 8. Consider the function f(x) = x10 . a. Use analytical methods to nd the exact value of f '(l) . b. Use the forward difference quotient to approximate f '(1) using values of h : 102, 103, and 10". What do you observe? c. Compute approximations to f'(l) using h = 10", for n = 5, 6, 7, ..., 15. What do you observe? In Step 8c, you should nd that for small enough values of h, the approximations to f '(I) eventually are 0, which is clearly a bad estimate. Here is why this error occurs. Suppose h = "T". The calculator rounds f(1+10'")-f(1) f (1 + 10714) to ] and therefore the forward difference quotient becomes 14 10 , which is 11 10*14 d. The remedy to rounding errors in this situation is to use smallibut not too smallivalues of h. Based on the approximations computed in parts (b) and (c), what is a good approximation to f'(l)? estimated to equal or 0