Question Description
I'm working on a calculus project and need an explanation and answer to help me learn.
Unformatted Attachment Preview
1. INTRODUCTION
This project focuses on ideas and concepts discussed in section 3.5 of your text. This section dis-
cusses the idea of linear approximations. A summary of the ideas:
(1) Suppose f(x) is differentiable at x = a. Then f'(a) exists and is the slope of the tangent
line to f(x) at x = a. The tangent line is given by
L(x) = f(a) + f'(a)(x – a).
=
(2) L(x) is called the Linear Approximation of f(x). This means that near x = a, we have
that
f(x) = L(x).
Note also that L(a) = f(a).
Review
vtbook on this
ma
omnlete the project.
wing
ical and eco-
omnical
WHAT TO HAND IN:
(1) The solutions to the following problems.
(2) Each problem should be clearly stated along with its solution.
(3) Please do not treat this project like another homework set. I expect for each problem a
comprehensive solution and clearly written conclusions, observations and interpretations.
Your project should read like a report.
(4) Make sure your name is written CLEARLY at the top of the page.
Due Date This project is due Nov 11 at class time.
Date: October 25, 2021.
1
2
PROF. LAZAROS KIKAS DEPARTMENT OF MATHEMATICS UNIVERSITY OF DETROIT MERCY
2. PROBLEMS
(1) (a) Find the linear approximation of tan x at x = - A
(b) Use part a to estimate tan(6 +0.02).
(c) Compute the percent error using
error
percenterror =
x 100.
actualvalue
where the error = estimatedvalue - actualvalue and actualvalue = 1.0408.
(2) Consider the curve y3 + 3xy = 7. Using implicit differentiation find dy and find the equa-
tion of the tangent line, L(x), to the curve at the point (2,1). Use L(x) to estimate the
y-coordinate of the point on the curve where x = 2.1.
=
(3) (a) Using linear approximation show that for any real number k, (1+Ax)k ~1+kAx for
small Ar. (hint: Find the linear approximation of f(x) = xk at x = = 1.)
(b) Compute (1.02)0.7 and (1.0001)37.
(c) Show that (8.24)1/3 ~ 2.02.
(4) Which is larger: V4.1 – V4 or V9.1 – V9? Use linear approximations to explain your
answers. Be sure to be clear, and use complete sentences.
(5) The data from US Department of Transportation shows that the number of vehicles regis-
tered in US was approximately 255,876,820 in 2013, and 260,350,940 in 2014. Using linear
approximation estimate the number of vehicles registered in the United States in 2016.
Remark: For Problem 5, do not just give me a number. Tell me what you are assuming
and why. Tell me your linear model, and use the model to predict. Explain your solution
using complete sentences and the neccessary math.
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