A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Section 3.5 Linear Approximation and Newton's Method

Solve the following problems.

1. The circumference of a sphere is measured to be 24 cm, with a possible error of 0.25 cm. Use the differential \(dV\) to estimate the maximum error in the calculated volume \(V\text{.}\)

2. The area \(A\) of a square of side length \(s\) is \(A=s^2\text{.}\) Suppose \(s\) increases by an amount \(\Delta s=ds\text{.}\)

   1. Draw a square and then illustrate the quantity \(dA\) on your diagram.

   2. If \(dA\) is used to approximate \(\Delta A\text{,}\) illustrate the error of approximation on the same diagram.

3. 1. Find the linear approximation to the function \(f(x)=\sqrt{(x+4)^3}\) at \(a=0\text{.}\)

   2. Use this approximation to estimate the number \(\sqrt{(3.95)^3}\text{.}\) Is your estimate an overestimate or an underestimate? (Hint: What is the concavity of the function \(f(x)\text{?}\))

4. 1. Use linear approximation to estimate \(\sqrt[3]{65}\text{.}\)

   2. Use concavity to state if your estimate in (a) is greater than or less than the exact value of \(\sqrt[3]{65}\text{.}\) Explain.

5. Use linear approximation to estimate the value of \(\sqrt[3]{26^2}\text{.}\) Express your answer a single fraction (for example, \(\ds \frac{16}{729}\)).

6. Use the linear approximation to approximate \((63)^{2/3}\text{.}\) Then use differentials to estimate the error.

7. Use linear approximation to estimate the value of \(\sqrt{80}\text{.}\)

8. Assume that \(f\) is function such that \(f(5)=2\) and \(f^\prime(5)=4\text{.}\) Using a linear approximation to \(f\) near \(x=5\text{,}\) find an approximation to \(f(4.9)\text{.}\)

9. Suppose that we don't have a formula for \(g(x)\) but we know that \(g(2)=-4\) and \(g^\prime (x)=\sqrt{x^2+5}\) for all \(x\text{.}\)

   1. Use linear approximation to estimate \(g(2.05)\text{.}\)

   2. Is your estimate in part (a) larger or smaller than the actual value? Explain.

10. 1. Find a linear approximation for the function \(f(x)=\sqrt{1-x}\) valid for \(x\) close to \(0\text{.}\)

    2. Use your answer to find an approximate value for \(\sqrt{0.9}\text{.}\)

    3. Find the tangent line to the graph of \(f(x)=\sqrt{1-x}\) at \(x=0\text{.}\)

    4. Sketch a graph to illustrate the relationship between \(f(x)=\sqrt{1-x}\) and its linear approximation near \(x=0\text{.}\)

11. Find the linear approximation of the function \(f(x)=\sqrt{1+x}\) at \(a=3\text{,}\) and use it to estimate the value of \(\sqrt{5}\text{.}\) Use a sketch to explain if this is an overestimate or underestimate of the actual value.

12. Let \(f(x)=\sqrt{1+2x}\text{.}\)

    1. Find the linear approximation of \(f(x)\) at \(x=0\text{.}\)

    2. Use your answer to estimate the value of \(\sqrt{1.1}\text{.}\)

    3. Is your estimate an over- or under-estimate?

13. 1. Find a linear approximation to the function \(f(x)=\sqrt[3]{x+8}\) at \(a=0\text{.}\)

    2. Use this approximation to estimate the numbers \(\sqrt[3]{7.95}\) and \(\sqrt[3]{8.1}\text{.}\)

14. 1. Construct the linear approximation to \(f(x)=(1+x)^{100}\text{.}\)

    2. Use your approximation from (a) to estimate \((1.0003)^{100}\text{.}\)

    3. Is your estimate from (b) higher or lower than the true value? Explain.

15. 1. Find the equation of the tangent line to the graph of the function \(f(x)=\sqrt[3]{27+3x}\) at \(x=0\text{.}\)

    2. Use your answer to estimate a value of \(\sqrt[3]{30}\text{.}\)

    3. Draw a sketch to show how the graph of \(f\) and its tangent line behave around the point where \(x=0\) and the value of \(x\) where the value in part (b) is obtained.

16. Use linear approximation to estimate the value of \(\ln 0.9\text{.}\)

17. Use a linear approximation to estimate the value of \(\ds e^{-0.015}\text{.}\) Is your estimate too large or too small?

18. 1. Write the linear approximation for \(f(x)=\ln x\) around 1.

    2. Compute the approximated value for \(\exp (-0.1)\) using linear approximation.

19. Using the function \(f(x)=x^{1/3}\) and the technique of linear approximation, give an estimate for \(1001^{1/3}\text{.}\)

20. Let \(\ds f(x)=\sqrt{x}+\sqrt[5]{x}\text{.}\)

    1. Use linear approximation to determine which of the following is nearest the value of \(f(1.001)\text{:}\)

       \begin{equation*} \begin{array}{lllll} 2.0001\amp 2.0002\amp 2.0003\amp 2.0005\amp 2.0007\\ 2.001\amp 2.002\amp 2.003\amp 2.005\amp 2.007 \end{array} \end{equation*}

    2. At \(x=1\text{,}\) is \(f(x)\) concave up or concave down?

    3. Based on your answer above, is your estimate of \(f(1.001)\) too high or too low?

21. 1. Find the linear approximation of \(f(x)=\sin x\) about the point \(x=\pi /6\text{.}\)

    2. Explain why \(f\) satisfies the conditions of the Mean Value Theorem. Use the theorem to prove that \(\ds \sin x\leq \frac{1}{2}+(x-\frac{\pi }{6})\) on the interval \(\ds [\frac{\pi }{6},x]\) where \(\ds x>\frac{\pi }{6}\)

    3. Is the differential \(df\) larger or smaller than \(\Delta f\) from \(\ds x=\frac{\pi }{6}\) to \(\ds x=\frac{\pi }{2}\text{?}\) Do not perform any calculations. Use only the results in part (a) and (b) to explain your answer.

22. Suppose that the only information we have about a function \(f\) is that \(f(1)=5\) and that the graph of its derivative is as shown in Figure 3.5.1.

    

    1. Use a linear approximation to estimate \(f(0.9)\) and \(f(1.1)\text{.}\)

    2. Are your estimates in part (a) too large or too small?

23. Suppose that the only information we have about a function \(f\) is that \(f(1)=3\) and that the graph of its derivative is as shown in Figure 3.5.2.

    

    1. Use a linear approximation to estimate \(f(0.9)\) and \(f(1.1)\text{.}\)

    2. Are your estimates in part (a) too large or too small?

24. 1. State Newton's iterative formula that produces a sequence of approximations \(x_1,x_2, x_3, \ldots\) to a root of function \(f(x)\text{.}\)

    2. Find the positive root of the equation \(\cos x=x^2\) using Newton's method, correct to 3 decimal points, with the first approximation \(x_1=1\text{.}\)

25. 1. State Newton's iterative formula that produces a sequence of approximations \(x_0,x_1,x_2, \ldots\) to a solution of \(f(x)=0\text{,}\) assuming that \(x_0\) is given.

    2. Draw a labeled diagram showing an example of a function \(f(x)\) for which Newton's iterative formula fails to find a solution of \(f(x)=0\text{.}\) Mark on your diagram \(x_0\text{,}\) \(x_1\text{,}\) and \(x_2\text{.}\)

26. 1. Explain how you can use Newton's Method to approximate the value of \(\sqrt{5}\text{.}\)

    2. Explain which of the following choices is the best initial approximation when using Newton's Method as in (a):\(-1\text{,}\) \(0\text{,}\) or \(1\text{?}\)

    3. Find the fourth approximation \(x_4\) to the value of \(\sqrt{5}\) using Newton's Method with the initial approximation \(x_1\) you chose in (b).

27. Apply Newton's method to \(\ds f(x)=x^{1/3}\) with \(x_0=1\) and calculate \(x_1,x_2,x_3, x_4\text{.}\) Find a formula for \(|x_n|\text{.}\) What happens to \(|x_n|\) as \(n\to \infty\text{?}\) Draw a picture that shows what is going on.

28. 1. Find the Newton's method iteration formula to compute to estimate \(\sqrt[3]{68}\text{.}\)

    2. Provide an initial guess. Then explain, whether your initial guess will lead to an over or under estimate after the first iteration.

29. 1. Use linear approximation to estimate \(\sqrt[3]{26}\text{.}\)

    2. The value of \(\sqrt[3]{26}\) is approximately \(x_1=3\text{.}\) Use Newton's method to find a better approximation, \(x_2\text{,}\) to \(\sqrt[3]{26}\text{.}\)

30. This question concerns finding zeros of the function

    \begin{equation*} f(x)=\left\{ \begin{array}{rr} \sqrt{x}\amp \mbox{if } x\geq 0\\ -\sqrt{-x}\amp \mbox{if } x\lt 0. \end{array} \right. \end{equation*}

    1. If the initial approximation to the zero, for \(f(x)\) given above, is \(x_1\text{,}\) what formula does Newton's method give for the next approximation?

    2. The root of the equation \(f(x)=0\) is \(x=0\text{.}\) Explain why Newton's method fails to find the root no matter which initial approximation \(x_1\not=0\) is used. Illustrate your explanation with a sketch.

31. 1. Suppose \(k\) is a constant. Show that if we apply Newton's method to approximate the value of \(\sqrt[5]{k}\text{,}\) we get the following iterative formula:

       \begin{equation*} x_{n+1}=\frac{x_n}{5}\left( 4+\frac{k}{x_n^5}\right)\text{.} \end{equation*}

    2. If \(x_n=\sqrt[5]{k}\text{,}\) what is the value of \(x_{n+1}\text{?}\)

    3. Take \(x_1=2\) and use the formula in part (a) to find \(x_2\text{,}\) an estimate of the value of \(\sqrt[5]{20}\) that is correct to one decimal place.

32. Use Newton's method to find the second approximation \(x_2\) of \(\sqrt[5]{31}\) starting with the initial approximation \(x_0=2\text{.}\)

33. 1. Suppose \(x_0\) is an initial estimate in Newton's method applied to the function \(f(x)\text{.}\) Derive Newton's formula for \(x_1\text{,}\) namely

       \begin{equation*} x_1=x_0-\frac{f(x_0)}{f'(x_0)}\text{.} \end{equation*}

       Support your derivation with a sketch showing a function \(f(x)\text{,}\) with \(x_0\text{,}\) \(x_1\) and the line whose slope is \(f'(x_0)\) clearly labeled.

    2. Using one iteration of Newton's method with \(\ds x_0=\frac{\pi }{2}\) approximate the \(x\)-coordinate of the point where the function \(g(x)=\sin x\) crosses the line \(y=x\text{.}\)

34. The equation

    \begin{equation*} 8x^3-12x^2-22x+25=0 \end{equation*}

    has a solution near \(x_1=1\text{.}\) Use Newton's Method to find a better approximation \(x_2\) to this solution. Express your answer as a fraction.

35. The tangent line to the graph \(y=f(x)\) at the point \(A(2,-1)\) is given by \(y=-1+4(x-2)\text{.}\) It is also known that \(f^{\prime\prime}(2)=3\text{.}\)

    1. Assume that Newton's Method is used to solve the equation \(f(x)=0\) and \(x_0=2\) is the initial guess. Find the next approximation, \(x_1\text{,}\) to the solution.

    2. Assume that Newton's Method is used to find a critical point for \(f\) and that \(x_0=2\) is the initial guess. Find the next approximation, \(x_1\text{,}\) to the critical point.

36. 1. Apply Newton's method to the equation \(\ds \frac{1}{x}-a=0\) to derive the following algorithm for finding reciprocals:

       \begin{equation*} x_{n+1}=2x_n-ax_n^2\text{.} \end{equation*}

    2. Use the algorithm from part (a) to calculate \(\ds \frac{1}{1.128}\) correct to three decimal places, starting with the first approximation \(x_1=1\text{.}\)

37. 1. Apply Newton's method to the equation \(\ds x^2-a=0\) to derive the following algorithm for the roots:

       \begin{equation*} x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)\text{.} \end{equation*}

    2. Approximate \(\sqrt{2}\) by taking \(x_1=2\) and calculating \(x_2\text{.}\)

38. 1. State the formula for the linearization of \(f\) at \(a\text{,}\)

    2. Using linear approximation, approximate \(\sqrt[4]{81.1}\text{.}\)

    3. Approximate \(\sqrt[4]{81.1}\) using one iteration of Newton's method.

39. You seek the approximate value of \(x\) which is near 1.8 for which \(\ds \sin x=\frac{x}{2}\text{.}\) Your first guess is that \(x\approx x_1=\frac{\pi }{2}\text{.}\) Use one iteration of Newton's method to find a better approximation to \(x\text{.}\) Simplify your answer as far as possible.

40. 1. For the function \(f(x)=x^3-3x+5\) use the Intermediate Value Theorem, and any other tools you need to determine intervals of length 1 each of which contains a root of \(f\text{.}\)

    2. Pick one of the intervals found in part (a). Choose the left endpoint of this interval to be \(x_0\text{.}\) Now, use this as a starting value to find two new iterations to the root of \(f\) by using Newton's method. Determine from these whether Newton's method is working. Justify your answer carefully.

41. Let \(f(x)=x^3+3x+1\text{.}\)

    1. Show that \(f\) has at least one root in the interval \(\ds \left( -\frac{1}{2},0\right)\text{.}\) Explain your reasoning.

    2. Use Newton's method to approximate the root that lies in the interval \(\ds \left( -\frac{1}{2},0\right)\text{.}\) Stop when the next iteration agrees with the previous one to two decimal places.

42. In this question we investigate the solution of the equation \(\ln x=-x^2+3\) on the interval \([1,3]\text{.}\)

    1. Explain why you know the equation has at least one solution on \([1,3]\text{.}\)

    2. Show that the equation has exactly one solution on \([1,3]\text{.}\)

    3. Use Newton's Method to approximate the solution of the equation by starting with \(x_1=1\) and finding \(x_2\text{.}\)

43. In this question we investigate the positive solution of the equation \(x^2+x=5-\ln x\text{.}\)

    1. Explain why you know the equation has at least one positive solution.

    2. Show that the equation has exactly one positive solution.

    3. Use Newton's Method to approximate the solution of the equation by starting with \(x_1=1\) and finding \(x_2\text{.}\)

44. In this question we investigate the solution of the equation \(2x=\cos x\text{.}\)

    1. Explain why you know the equation has at least one solution.

    2. Show that the equation has exactly one solution.

    3. Use Newton's Method to approximate the solution of the equation by starting with \(x_1=0\) and finding \(x_2\text{.}\)

45. In this question we investigate the solution of the equation \(2x-1=\sin x\text{.}\)

    1. Explain why you know the equation has at least one solution.

    2. Show that the equation has exactly one solution.

    3. Use Newton's Method to approximate the solution of the equation by starting with \(x_1=0\) and finding \(x_2\text{.}\)

46. In this question we investigate the positive solution of the equation \(e^x=2\cos x\text{.}\)

    1. Explain why you know the equation has at least one positive solution.

    2. Show that the equation has exactly one positive solution.

    3. Use Newton's Method to approximate the solution of the equation by starting with \(x_1=0\) and finding \(x_2\text{.}\)

47. Consider the equation

    \begin{equation*} x^6-x-1=0\text{.} \end{equation*}

    1. Apply the Intermediate Value Theorem to the function \(f(x)=x^6-x-1\) to prove that the given equation has a root greater than 1. Make sure that you justify why the function \(f\) is continuous on its domain.

    2. Use the derivative of the function \(f(x)=x^6-x-1\) to prove that the given equation has only one root greater than 1. Call that root \(a\text{.}\) Show all your work. Clearly explain your reasoning.

    3. State Newton's Method.

    4. Use Newton's Method with the initial approximation \(x_1=1\) to find \(x_2\) and \(x_3\text{,}\) the second and the third approximations to the root \(a\) of the equation \(x^6-x-1=0\text{.}\) You may use your calculator to find those values. Show all your work. Clearly explain your reasoning.

    5. WolframAlfa gives \(a\approx 1.13472\text{.}\) Use your calculator to evaluate the number \(|x_3 -1.13472|\text{.}\) Are you satisfied with your approximation. Why yes or why not?

48. 1. State Rolle's theorem.

    2. Use Rolle's theorem to prove that \(f(x)\) has a critical point in \([0,1]\) where

       \begin{equation*} f(x)=\sin\left(\frac{\pi x}{2}\right) -x^2\text{.} \end{equation*}

    3. Set up the Newton's method iteration formula (\(x_{n+1}\) in terms of \(x_n\)) to approximate the critical point You do not need to simplify.

49. 1. State the Mean Value Theorem.

    2. Using the Mean Value Theorem, prove that \(f(x)\) has a critical point in \([0,1]\) where

       \begin{equation*} f(x)=\cos\left(\frac{\pi x}{2}\right) +x\text{.} \end{equation*}

    3. Set up the Newton's method iteration formula to approximate the critical point You do not need to simplify.

50. 1. State the Intermediate Value Theorem.

    2. State the Mean Value Theorem.

    3. Use the Intermediate Value Theorem and the Mean Value Theorem to show that the equation \(1+2x+x^3+4x^5=0\) has exactly one real root.

51. A function \(h(x)\) is said to have a fixed point at \(x=c\) if \(h(c)=c\text{.}\) Suppose that the domain and range of a function \(f(x)\) are both the interval \([0,1]\) and that \(f\) is continuous on this domain, with \(f(0)\not= 0\) and \(f(1)\not=1\text{.}\)

    1. Prove that \(f\) has at least one fixed point. That is, prove that \(f(c)=c\) for some \(c\in(0,1)\text{.}\)

    2. Suppose that \(f^\prime(x)\lt 1\) for all \(x\in(0,1)\text{.}\) Prove that \(f\) has exactly one fixed point in \([0,1]\text{.}\)

    3. Use Newton's method to determine an iteration formula for the fixed point \(x=c\text{.}\)