Exam3
- Page ID
- 219472
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)MATH 105 PRACTICE MIDTERM III
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a counter example. If true, explain why.
A) If f(x) is a positive continuous function such that
\( \int_1^3 f(x)dx = 4 \)
then
\( \int_0^5 f(x)dx\)
cannot be equal to 3.
Solution
B) If f(x) is a differentiable function such that the equation of the tangent line at
x = 2
is
1 1
y = x -
2 2
and if x = 2 is the first guess in Newton’s method, then x = 1 is the second guess.
Solution
C) If f(x) and g(x) are continuous functions on [a,b], then
\( \int_a^b f(x)g(x)dx =(\int_a^b f(x)dx) (\int_a^b g(x)dx) \)
PROBLEM 2 Evaluate the following integrals:
A. \( \int (sin(x) - \sqrt{x} + 2x^2 - 3)dx\)
Solution
B. \( \int_2^3 \frac{x}{(1 - x)^3}dx\)
Solution
C. \( \int csc(2x)cot(2x)dx\)
Solution
PROBLEM 3 Use Riemann Sums to find the area of the region below the curve y = 9 - x2, above the x-axis, and between x = 1 and x = 3.
PROBLEM 4
Let
\( \int_3^{sin(x)} cos(t^2)dt\)
Find F'(x)
PROBLEM 5
You are the owner of Tahoe Winter Wear and need to determine the best price to sell your most popular winter jacket. Your cost for selling the jackets is
C = 50 + 20x
Where x is the amount to jackets that you sell. Your research shows that the relationship between price, p, and the number of jackets that you can sell, x, is
p = 300 – 10x
How much should you charge for your jacket in order to maximize profit?
PROBLEM 6
You are manufacturing a square computer chip. Your machine can construct the square with side length 0.4 
0.0002 cm. Use differentials to approximate the maximum percent error in the area of the chip.
PROBLEM 7
Let f(x) = x3 + x + 4
-
Prove that f(x) has an inverse function.
Solution -
Let g(x) be the inverse of f(x). Find g'(4) .
Solution