MATH1013 Calculus I

Tutorial 8 Linear Approximation

(A) Linear Approximation

1) Linearization of a function

Example 1

2) Differentials

Example 2

3) Application

Example 3 The radius of a sphere was measured and found to be 21 cm with a possible error in

measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to

compute the volume of the sphere?

(B) Newton’s Method

(1) Newton’s Method of finding Roots

PROCEDURE Newton’s Method for Approximating Roots of 𝒇(𝒙) = 𝟎

1. Choose an initial approximation 𝑥0 as close to a root as possible.

2. For n = 0, 1, 2, …

𝒙𝒏+𝟏 = 𝒙𝒏 −𝒇(𝒙𝒏)

𝒇′(𝒙𝒏) , provided 𝑓′(𝑥𝑛) ≠ 0.

3. End the calculations when a termination condition is met.

Example 5

(C) Maximum and Minimum Value

1) Increasing and Decreasing Function

2) Test for Intervals of Increase and Decrease

3) Absolute Maximum and Absolute Minimum

implies 𝒅𝒚 = 𝒇′(𝒙)𝒅𝒙

4) Extreme Value Theorem

5) Local Maximum and Local Minimum

6) Local Extreme Point Theorem

7) Use the 1st Derivative of 𝒇(𝒙) to find the Extreme Values

8) The 2nd

Derivative Test for Extremums

9) Test for Local Maximum and Local Minimum

(a) For the curve 𝒚 = 𝒇(𝒙). If both 𝒇(𝒙) and 𝒇′(𝒙) are differentiable at 𝒙 = 𝒙𝟎 , and 𝒇′(𝒙𝟎) = 𝟎 ,

𝒇"(𝒙𝟎) < 𝟎 , then (𝒙𝟎 , 𝒇(𝒙𝟎) is a local maximum point

(b) For the curve 𝒚 = 𝒇(𝒙) . If both 𝒇(𝒙) and 𝒇′(𝒙) are differentiable at 𝒙 = 𝒙𝟎 , and 𝒇′(𝒙𝟎) = 𝟎 ,

𝒇"(𝒙𝟎) > 𝟎 , then (𝒙𝟎 , 𝒇(𝒙𝟎) is a local minimum point

Example 6

Exercises

Linear Approximation

1) Find the linearization of the following functions at the spacified point 𝒂.

(a) 𝒇(𝒙) = 𝒙𝟒 + 𝟑𝒙𝟐 , 𝒂 = −𝟏

(b) 𝒇(𝒙) = 𝒄𝒐𝒔𝒙 , 𝒂 =𝝅

𝟐 .

2) Find the linear approximation of the function 𝒇(𝒙) = √𝟏 − 𝒙 at 𝒂 = 𝟎 and use it to

approximate the number √𝟎. 𝟗 and √𝟎. 𝟗𝟗 . Illustrate by graphing 𝒇 and the tangent line.

3) Verify the linear aproximation 𝟏

(𝟏+𝟐𝒙)𝟒 ≈ 𝟏 − 𝟖𝒙 at 0. Then determine the value of x for which

the linear aproximation is accurate to within 0.1. 4) Use a linear aproximation (or differentials ) to estimate the following numbers.

(a) (𝟐. 𝟎𝟎𝟎𝟏)𝟓 (b) (𝟖. 𝟎𝟔) 𝟐

𝟑

(c) 𝒕𝒂𝒏 𝟒𝟒𝟎 (d) 𝒍𝒏 (𝟏. 𝟎𝟓)

(e) 𝟏

√𝟏𝟏𝟗 (f) 𝒆𝟎.𝟎𝟔

For Questions 5 – 6

(a) Write the equation of the line that represents the linear approximation to the

following functions at the given point 𝒂.

(b) Graph the function and the linear approximation at 𝒂. (c) Use the linear approximation to estimate the given function value.

(d) Compute the percent error in your approximation, 𝟏𝟎𝟎 ∙|𝐚𝐩𝐩𝐫𝐨𝐱 – 𝐞𝐱𝐚𝐜𝐭|

|𝐞𝐱𝐚𝐜𝐭| , where the

exact value is given by a calculator.

5) 𝒇(𝒙) = (𝟖 + 𝒙)−𝟏

𝟑 ; 𝒂 = 𝟎 ; 𝒇(−𝟎. 𝟏) .

6) 𝒇(𝒙) = √𝒙𝟒

; 𝒂 = 𝟖𝟏 ; 𝒇(𝟖𝟓) . 7) (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell

with height 𝒉 , inner radius 𝒓 , and thickness ∆𝒓 . (b) What is the error involved in using the formula from part (a)?

8) If a current 𝑰 passes through a resistor with resistance 𝑹 , Ohm’s Law states that the voltage

drop is 𝑽 = 𝑰𝑹 . If 𝑽 is constant and 𝑹 is measured with a certain error, use differentials to

show that the relative error in calculating 𝑰 is approximately the same (in magnitude) as the

relative error in 𝑹 .

Newton’s method

Q9 - 10 ) Finding roots with Newton’s method

Use a calculator or program to compute the first 10 iterations of Newton’s method when they are applied to the following functions with the given initial approximation. Make a table similar to show the steps.

9) 𝒇(𝒙) = 𝒔𝒊𝒏𝒙 + 𝒙 − 𝟏 ; 𝒙𝟎 = 𝟏. 𝟓 (Briggs 4.8#11)

10) 𝒇(𝒙) = 𝒍𝒏(𝒙 + 𝟏) − 𝟏 ; 𝒙𝟎 = 𝟏. 𝟕 (Briggs 4.8#14)

Q11 - 13) Finding intersection points

Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.

11) 𝒚 = 𝒙𝟑 and 𝒚 = 𝒙𝟐 + 𝟏 (Briggs 4.8#18)

12) 𝒚 = 𝟒√𝒙 and 𝒚 = 𝒙𝟐 + 𝟏 (Briggs 4.8#19)

13) 𝒚 = 𝒍𝒏𝒙 and 𝒚 = 𝒙𝟑 − 𝟐 (Briggs 4.8#20)

14) Residuals and errors (Briggs 4.8#39)

Approximate the root of 𝑓(𝑥) = 𝑥10 at 𝑥 = 0 using Newton’s method with an initial approximation of 𝑥0 = 0.5. Make a table showing the first 10 approximations, the error in these approximations

(which is |𝑥𝑛 − 0| = |𝑥𝑛|) , and the residual of these approximations (which is 𝑓(𝑥𝑛) ). Comment on the relative size of the errors and the residuals, and give an explanation.

Maximum and Minimum Value

15) Prove that the function 𝒇(𝒙) = 𝒙𝟏𝟎𝟏 + 𝒙𝟓𝟏 + 𝒙 + 𝟏 has neither a local maximum nor a local minimum. (4.1#75)

16) Find the absolute maximum and absolute minimum values of 𝒇 on the given interval.

(a) 𝒇(𝒙) = 𝟑𝒙𝟐 − 𝟏𝟐𝒙 + 𝟓 , [𝟎, 𝟑] (4.1#47) (6ed)

(b) 𝒇(𝒙) =𝒙

𝒙𝟐+𝟏 , [𝟎, 𝟐] (4.1#53) (6ed)

(c) 𝒇(𝒙) = 𝒙𝒆−𝒙𝟐

𝟖 , [−𝟏, 𝟒] (4.1#53)

(d) 𝒇(𝒙) = 𝒍𝒏(𝒙𝟐 + 𝒙 + 𝟏) , [−𝟏, 𝟏] (4.1#61)

17) L’H𝒐pital Rule

If 𝒇’ is continuous, 𝒇(𝟐) = 𝟎 , and 𝒇’(𝟐) = 𝟕 , evaluate 𝐥𝐢𝐦𝒙→𝟎𝒇(𝟐+𝟑𝒙)+𝒇(𝟐+𝟓𝒙)

𝒙 .

Optimization Problems

18) A model used for the yield 𝒀 of an agricultural crop as a function of the nitrogen level 𝑵 in

the soil (measured in appropriate units) is 𝒀 =𝒌𝑵

𝟏+𝑵𝟐 , where 𝑲 is a positive constant.

What nitrogen level gives the best yield? (4.7#9) 19) A farmer wants to fence an area of 1.5 milion square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimiz the cost of the fence? (4.7#13)

20) Find the point on the line 𝒚 = 𝟒𝒙 + 𝟕 that is closest to the origin. (4.7#7 )(6ed)

21) A right circular cylinder is inscribed in a sphere of radius 𝒓. Find the largest possible surface area of such a cylinder. (4.7#31)

22) A cone-shaped drinking cup is made from a circular piece of

paper of radius 𝑹 by cutting out a sector and joining the edges 𝑪𝑨 and 𝑪𝑩. Find the maximum capacity of such a cup. (4.7#39)

23) A cone with height 𝒉 is inscribed in a larger cone with height 𝑯 so that its vertex is at the center of the base of the larger cone. Show

that the inner cone has maximum volume when 𝒉 =𝟏

𝟑𝑯. (4.7#41)

24) A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) a minimum? (4.7#33) (6ed)

25) A baseball team plays in a statium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (4.7#59) (a) Find the demand function, assuming that it is linear. (b) How shoule ticket prices be set to maximize revenue?

26) The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In

other words, how would you choose 𝒙 to minize 𝒚? (4.7#69)

27) Let 𝒗𝟏 be the velocity of light in air and 𝒗𝟐 the velocity of light in water. According to Fermat’s Principle, a ray of

light will travel from a point 𝑨 in the air to a Point 𝑩 in the water by a path 𝑨𝑪𝑩 that minimizes the time taken. Show that 𝒔𝒊𝒏 𝜽𝟏

𝒔𝒊𝒏 𝜽𝟐=

𝒗𝟏

𝒗𝟐, where 𝜽𝟏 (the angle of incidence) and 𝜽𝟐 (the angle

of refraction) are shown. This eqution is known as Snell’s Law. (4.7#67)

28) An observer stands at a point 𝑷, one unit away from a track.

Two runners start at the point 𝑺 in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum

value of the observer’s angle of sight 𝜽 between the runners. [Hint: Maximize 𝒕𝒂𝒏 𝜽 ] (4.7#71)