While the rules of differentiation allow us to compute the derivative of just about any function,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
While the rules of differentiation allow us to compute the derivative of just about any function, there are practical situations in which these rules cannot be used. For example, in some applications, a relationship between two variables may be given as a set of data points, but not as a formula. In situations like this, the rate of change of one variable with respect to the other (that is, the derivative) might be needed, but the rules do not apply to sets of data. This project focuses on methods for approximating the derivative of a function at a particular point. Backward and Forward Difference Quotients Assuming the limit exists, the definition of the derivative f'(a) = lim 4-0 f'(a)= f(a+h)-f(a) h (1) for h near 0. If h> 0, then (1) is referred to as a forward difference quotient and if h<0. (1) is a backward difference quotient. The geometry of these formulas is shown in Figure 1. (a.fla)) 0.1 0.01 0.001 0.0001 -y-fux) (a+h) a Backward difference: <0 f(a)- X √4+h-2 h fla+ h)-f(a) h ƒ(4+h)-f(4)_ √√4+h-2 h h Error Figure 1 1. why do you think (1) is called the forward difference quotient if h> 0 and a backward difference quotient ifh < 0? 2. Let /(x)=√x. a. Find the exact value of ƒ'(4). b. By equation (1), f'(4)= √4+/-2 h behaves as h approaches 0. h 3. The accuracy of an approximation is given by f(a+h)-f(a) h Table 1 implies that (a, f(a)) for values of h near 0. Complete columns 2 and 5 of Table 1 and describe how -0.1 -0.01 -0.001 -0.0001 y=foo Forward difference: >0 X Therefore we estimate f'(4) by calculating √4+h-2 √4+h-2 Error h Error = exact value-approximate value. Use the exact value of f'(4) in part (a) to complete columns 3 and 6 in Table 1. Describe the behavior of the errors as / approaches 0. Centered Difference Quotients Another formula is used to approximate the derivative of a function at a point is the centered difference quotient (CDQ) f'(a)=f(a+h)-f(a-h) 2h (2) 3. Again consider f(x)=√x. a. Draw the graph of f near the point (4, 2) and let h = in the centered difference quotient. Compute and draw the line whose slope is computed by the centered difference quotient and explain why the formula approximates f'(4). Use the CDQ formula to approximate f'(4) by completing Table 2. Give answers accurate to nine decimal places. Approximation b. h Error 0.1 0.01 0.001 0.0001 Table 2 c. Explain why it is not necessary to use negative values of h in Table2. d. In a sentence, compare the accuracy of the derivative estimates in part (b) with those found in Questions 2 and 3. 4. Use the CDQ formula and a table similar to Table 2 to find a good approximation to f'(0) for f (x) = (1 +x)-¹. Give answers accurate to at least six decimal places. You may use scientific notation to express the error. 5. Use the CDQ formula and a table similar to Table 2 to find a good approximation to f'() for f (x) = sin x. Give answers accurate to at least six decimal places. You may use scientific notation to express the error. 6. Table 3 gives the distance fr) fallen by a smokejumper / seconds after she opens her chute. a. Use the forward difference quotient (Formula 1) with=05 to estimate the velocity of the skydiver at f= 2 s. b. Repeat part (a) using the centered difference quotient (Formula 2). 1 (seconds) Ar) (feet) 0 0 4 15 33 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Table 3 55 81 109 138 169 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. 7. Consider the function f(x) = x¹0, a. Use analytical methods (i.e. derivative rules) to find the exact value of f'(1). b. Use the forward difference quotient to approximate f'(1) using values h= 102, 103, and 10 Then in a sentence, answer what do you observe? c. Compute approximations to f'(1) using h=10 for n= 5,6,7,... 15. Then in a sentence, answer what do you observe? In Stop Se, you should find that for small mough values of A, the approximations tof'(1) eventually are 0, which is clearly a bad estimate. Here is why this error occurs. Suppose A-1014 The calculator rounds /(1+ 10-14) to 1 and therefore the forward difference quotient becomes ), which is estimated to equal or 0.) 10-1 d. The remedy to rounding errors in this situation is to use small but not too small-values of . Based on the approximations computed in parts (b) and (c), what is a good approximation to f'(1)? While the rules of differentiation allow us to compute the derivative of just about any function, there are practical situations in which these rules cannot be used. For example, in some applications, a relationship between two variables may be given as a set of data points, but not as a formula. In situations like this, the rate of change of one variable with respect to the other (that is, the derivative) might be needed, but the rules do not apply to sets of data. This project focuses on methods for approximating the derivative of a function at a particular point. Backward and Forward Difference Quotients Assuming the limit exists, the definition of the derivative f'(a) = lim 4-0 f'(a)= f(a+h)-f(a) h (1) for h near 0. If h> 0, then (1) is referred to as a forward difference quotient and if h<0. (1) is a backward difference quotient. The geometry of these formulas is shown in Figure 1. (a.fla)) 0.1 0.01 0.001 0.0001 -y-fux) (a+h) a Backward difference: <0 f(a)- X √4+h-2 h fla+ h)-f(a) h ƒ(4+h)-f(4)_ √√4+h-2 h h Error Figure 1 1. why do you think (1) is called the forward difference quotient if h> 0 and a backward difference quotient ifh < 0? 2. Let /(x)=√x. a. Find the exact value of ƒ'(4). b. By equation (1), f'(4)= √4+/-2 h behaves as h approaches 0. h 3. The accuracy of an approximation is given by f(a+h)-f(a) h Table 1 implies that (a, f(a)) for values of h near 0. Complete columns 2 and 5 of Table 1 and describe how -0.1 -0.01 -0.001 -0.0001 y=foo Forward difference: >0 X Therefore we estimate f'(4) by calculating √4+h-2 √4+h-2 Error h Error = exact value-approximate value. Use the exact value of f'(4) in part (a) to complete columns 3 and 6 in Table 1. Describe the behavior of the errors as / approaches 0. Centered Difference Quotients Another formula is used to approximate the derivative of a function at a point is the centered difference quotient (CDQ) f'(a)=f(a+h)-f(a-h) 2h (2) 3. Again consider f(x)=√x. a. Draw the graph of f near the point (4, 2) and let h = in the centered difference quotient. Compute and draw the line whose slope is computed by the centered difference quotient and explain why the formula approximates f'(4). Use the CDQ formula to approximate f'(4) by completing Table 2. Give answers accurate to nine decimal places. Approximation b. h Error 0.1 0.01 0.001 0.0001 Table 2 c. Explain why it is not necessary to use negative values of h in Table2. d. In a sentence, compare the accuracy of the derivative estimates in part (b) with those found in Questions 2 and 3. 4. Use the CDQ formula and a table similar to Table 2 to find a good approximation to f'(0) for f (x) = (1 +x)-¹. Give answers accurate to at least six decimal places. You may use scientific notation to express the error. 5. Use the CDQ formula and a table similar to Table 2 to find a good approximation to f'() for f (x) = sin x. Give answers accurate to at least six decimal places. You may use scientific notation to express the error. 6. Table 3 gives the distance fr) fallen by a smokejumper / seconds after she opens her chute. a. Use the forward difference quotient (Formula 1) with=05 to estimate the velocity of the skydiver at f= 2 s. b. Repeat part (a) using the centered difference quotient (Formula 2). 1 (seconds) Ar) (feet) 0 0 4 15 33 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Table 3 55 81 109 138 169 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. 7. Consider the function f(x) = x¹0, a. Use analytical methods (i.e. derivative rules) to find the exact value of f'(1). b. Use the forward difference quotient to approximate f'(1) using values h= 102, 103, and 10 Then in a sentence, answer what do you observe? c. Compute approximations to f'(1) using h=10 for n= 5,6,7,... 15. Then in a sentence, answer what do you observe? In Stop Se, you should find that for small mough values of A, the approximations tof'(1) eventually are 0, which is clearly a bad estimate. Here is why this error occurs. Suppose A-1014 The calculator rounds /(1+ 10-14) to 1 and therefore the forward difference quotient becomes ), which is estimated to equal or 0.) 10-1 d. The remedy to rounding errors in this situation is to use small but not too small-values of . Based on the approximations computed in parts (b) and (c), what is a good approximation to f'(1)?
Expert Answer:
Answer rating: 100% (QA)
The forward difference quotient is called forward because it uses a positive value of h which means it considers the points ahead or forward of the gi... View the full answer
Related Book For
Computer Organization and Design The Hardware Software Interface
ISBN: 978-0124077263
5th edition
Authors: David A. Patterson, John L. Hennessy
Posted Date:
Students also viewed these accounting questions
-
Directions Using the tax software, complete the tax retur, including Form 1040 and all appropri ate forms, schedules, or worksheets Answer the questions following the scenario Note: When entering...
-
A scatter plot can reveal a relationship between two indicators. Construct a scatter plot of annual data beginning in 1959 for inflation and money growth. Measure these as the percent change from a...
-
You can have a strong relationship between two variables but still, have a low correlation coefficient when: a) the relationship is not linear b) variables are truncated c) both a and b d) neither a...
-
In Exercises 8182, graph each linear function. 3x - 4f(x) - 6 = 0
-
Describe the differences between Black Belts, Green Belts, and Master Black Belts in a Six Sigma program.
-
A company's Cash account shows a balance of 3 , 4 5 0 at the end of the month. Comparing the company's Cash account with the monthly bank statement reveals several additional cash transactions such...
-
Let \(X_{t}, t \geq 0\), be defined as \[X_{t}=\left\{B_{t} \mid B_{t} \geq 0 ight\}, \quad \forall t>0\] that is, the process has the paths of the Brownian motion conditioned by the current value...
-
The following preliminary unadjusted trial balance of Ranger Co., a sports ticket agency, does not balance: When the ledger and other records are reviewed, you discover the following: (1) the debits...
-
Besides possibly paying the premiums for a medical insurance plan, the insured is almost always responsible for making other, out-of-pocket payments. This practice helps keep overall costs down for...
-
1. Is Facebook's mission statement market oriented? Explain. 2. How is Facebook's strategy driven by its mission? 3. Is it wise for Facebook to give away it technologies for free? Why or why not? 4....
-
Additional paid-in capital is most likely to appear in the balance sheet of a corporation that: a) has par value stock. b) has no-par value stock. c) has issued stock at different dates. d) has...
-
Explain why it was so complicated to take out a mortgage loan to buy a house from a commercial bank in the 1870s.
-
Explain why asymmetric information problems do not prevent money market participants from buying and selling money market instruments.
-
Consider a 30-year fixed-rate mortgage for $200,000 at a nominal rate of 8%. A mortgage broker closes this deal on August 30. If the broker wants to sell this mortgage to a bank on August 31 and...
-
Which fringe benefits are income taxable? A. Employee achievement awards. B. On-premise athletic facilities. C. Prepaid legal services. D. Occasional personal use of a company copying machine.
-
In which circumstances would it be appropriate to take out a second mortgage?
-
There is a bonus structure in place that gives salespeople a $1000 bonus for producing sales levels of at least $15,000 in a month and a $500 bonus for producing sales levels between $13,000 and...
-
Explain how the graph of each function can be obtained from the graph of y = 1/x or y = 1/x 2 . Then graph f and give the (a) Domain (b) Range. Determine the largest open intervals of the domain over...
-
Calculate 3.984375 10 -1 + (3.4375 10 -1 + 1.771 10 3 ) by hand, assuming each of the values are stored in the 16-bit half precision format described in Exercise 3.27 (and also described in the...
-
The following instruction is not included in the MIPS instruction set: rpt $t2, loop # if(R[rs]>0) R[rs]=R[rs]1, PC=PC+4+BranchAddr 1. If this instruction were to be implemented in the MIPS...
-
Provide the type and hexadecimal representation of following instruction: sw $t1, 32($t2)
-
a. Suppose that General Hospital has a current ratio of 0.5. Which of the following actions would improve (increase) this ratio? Use cash to pay off current liabilities. Collect some of the current...
-
What is the role of internal control in an organization?
-
What are the elements and principles of the COSO framework?
Study smarter with the SolutionInn App