math1013 calculus i tutorial 8 linear approximation (a
TRANSCRIPT
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) Newtonβs Method
(1) Newtonβs Method of finding Roots
PROCEDURE Newtonβs Method for Approximating Roots of π(π) = π
1. Choose an initial approximation π₯0 as close to a root as possible.
2. For n = 0, 1, 2, β¦
ππ+π = ππ βπ(ππ)
πβ²(ππ) , provided πβ²(π₯π) β 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 π π = πβ²(π)π π
4) Extreme Value Theorem
5) Local Maximum and Local Minimum
6) Local Extreme Point Theorem
7) Use the 1st Derivative of π(π) to find the Extreme Values
8) The 2nd
Derivative Test for Extremums
9) Test for Local Maximum and Local Minimum
(a) For the curve π = π(π). If both π(π) and πβ²(π) are differentiable at π = ππ , and πβ²(ππ) = π ,
π"(ππ) < π , then (ππ , π(ππ) is a local maximum point
(b) For the curve π = π(π) . If both π(π) and πβ²(π) are differentiable at π = ππ , and πβ²(ππ) = π ,
π"(ππ) > π , then (ππ , π(ππ) is a local minimum point
Example 6
Exercises
Linear Approximation
1) Find the linearization of the following functions at the spacified point π.
(a) π(π) = ππ + πππ , π = βπ
(b) π(π) = ππππ , π =π
π .
2) Find the linear approximation of the function π(π) = βπ β π at π = π and use it to
approximate the number βπ. π and βπ. ππ . Illustrate by graphing π and the tangent line.
3) Verify the linear aproximation π
(π+ππ)π β π β ππ 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) (π. ππππ)π (b) (π. ππ) π
π
(c) πππ πππ (d) ππ (π. ππ)
(e) π
βπππ (f) ππ.ππ
For Questions 5 β 6
(a) Write the equation of the line that represents the linear approximation to the
following functions at the given point π.
(b) Graph the function and the linear approximation at π. (c) Use the linear approximation to estimate the given function value.
(d) Compute the percent error in your approximation, πππ β|ππ©π©π«π¨π± β ππ±πππ|
|ππ±πππ| , where the
exact value is given by a calculator.
5) π(π) = (π + π)βπ
π ; π = π ; π(βπ. π) .
6) π(π) = βππ
; π = ππ ; π(ππ) . 7) (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell
with height π , inner radius π , and thickness βπ . (b) What is the error involved in using the formula from part (a)?
8) If a current π° passes through a resistor with resistance πΉ , Ohmβs Law states that the voltage
drop is π½ = π°πΉ . If π½ is constant and πΉ is measured with a certain error, use differentials to
show that the relative error in calculating π° is approximately the same (in magnitude) as the
relative error in πΉ .
Newtonβs method
Q9 - 10 ) Finding roots with Newtonβs method
Use a calculator or program to compute the first 10 iterations of Newtonβs method when they are applied to the following functions with the given initial approximation. Make a table similar to show the steps.
9) π(π) = ππππ + π β π ; ππ = π. π (Briggs 4.8#11)
10) π(π) = ππ(π + π) β π ; ππ = π. π (Briggs 4.8#14)
Q11 - 13) Finding intersection points
Use Newtonβs 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) π = ππ and π = ππ + π (Briggs 4.8#18)
12) π = πβπ and π = ππ + π (Briggs 4.8#19)
13) π = πππ and π = ππ β π (Briggs 4.8#20)
14) Residuals and errors (Briggs 4.8#39)
Approximate the root of π(π₯) = π₯10 at π₯ = 0 using Newtonβs method with an initial approximation of π₯0 = 0.5. Make a table showing the first 10 approximations, the error in these approximations
(which is |π₯π β 0| = |π₯π|) , and the residual of these approximations (which is π(π₯π) ). Comment on the relative size of the errors and the residuals, and give an explanation.
Maximum and Minimum Value
15) Prove that the function π(π) = ππππ + πππ + π + π has neither a local maximum nor a local minimum. (4.1#75)
16) Find the absolute maximum and absolute minimum values of π on the given interval.
(a) π(π) = πππ β πππ + π , [π, π] (4.1#47) (6ed)
(b) π(π) =π
ππ+π , [π, π] (4.1#53) (6ed)
(c) π(π) = ππβππ
π , [βπ, π] (4.1#53)
(d) π(π) = ππ(ππ + π + π) , [βπ, π] (4.1#61)
17) LβHπpital Rule
If πβ is continuous, π(π) = π , and πβ(π) = π , evaluate π₯π’π¦πβππ(π+ππ)+π(π+ππ)
π .
Optimization Problems
18) A model used for the yield π of an agricultural crop as a function of the nitrogen level π΅ in
the soil (measured in appropriate units) is π =ππ΅
π+π΅π , where π² 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 π = ππ + π that is closest to the origin. (4.7#7 )(6ed)
21) A right circular cylinder is inscribed in a sphere of radius π. 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 πΉ by cutting out a sector and joining the edges πͺπ¨ and πͺπ©. Find the maximum capacity of such a cup. (4.7#39)
23) A cone with height π is inscribed in a larger cone with height π― so that its vertex is at the center of the base of the larger cone. Show
that the inner cone has maximum volume when π =π
ππ―. (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 π to minize π? (4.7#69)
27) Let ππ be the velocity of light in air and ππ the velocity of light in water. According to Fermatβs Principle, a ray of
light will travel from a point π¨ in the air to a Point π© in the water by a path π¨πͺπ© that minimizes the time taken. Show that πππ π½π
πππ π½π=
ππ
ππ, where π½π (the angle of incidence) and π½π (the angle
of refraction) are shown. This eqution is known as Snellβs Law. (4.7#67)
28) An observer stands at a point π·, one unit away from a track.
Two runners start at the point πΊ in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum
value of the observerβs angle of sight π½ between the runners. [Hint: Maximize πππ π½ ] (4.7#71)