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Find an equation of the plane. The plane through the origin and perpendicular to the line $$ x=1-8 t \quad y=-1-7 t \quad z=4+2 t $$

Find an equation of the plane. The plane through the origin and perpendicular to the line $$ x=1-8 t \quad y=-1-7 t \quad z=4+2 t $$

Calculus: Early Transcendentals, Metric Edition

Vectors and the Geometry of Space

Equations of Lines and Planes

Determine whether the series is convergent or divergent. $\sum_{n=1}^{\infty} \frac{n}{n^{4}+1}$

Calculus: Early Transcendentals, Metric Edition

Sequences, Series, and Power Series

The Integral Test and Estimates of Sums

A $4-\mathrm{kN}$ force $\mathbf{P}$ is applied as shown to the piston of the engine system. Knowing that $A B=50 \mathrm{mm}$ and $B C=200 \mathrm{mm}$, determine the couple $\mathrm{M}$ required to maintain the equilibrium of the system when $(a) \theta=30^{\circ},(b) \theta=150^{\circ} .$ (Graph cannot copy)

A $4-\mathrm{kN}$ force $\mathbf{P}$ is applied as shown to the piston of the engine system. Knowing that $A B=50 \mathrm{mm}$ and $B C=200 \mathrm{mm}$, determine the couple $\mathrm{M}$ required to maintain the equilibrium of the system when $(a) \theta=30^{\circ},(b) \theta=150^{\circ} .$ (Graph cannot copy)

Vector Mechanics for Engineers: Statics and Dynamics

Method of Virtual Work

The Basic Method

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Calculus: Early Transcendentals

Functions and Models

Four Ways to Represent a Function

Questions asked

INSTANT ANSWER

Randy is attending film school and one of his projects involves researching the ratings given to movies by critics and viewers. He has selected a random sample of movies and has recorded the average critic score (X1) and the average viewer score (X2) for each movie. The scores were obtained from a website that collates movie reviews and ratings submitted by critics and viewers. Randy also recorded the genre (X3) of each movie. The data are stored in the file AssignmentData.RData in the data frame movies.df. c) [3 marks] For thrillers, test whether the population proportion of movies that have an average viewer score greater than 3.27 is less than 0.5. Clearly state your hypothe- ses, making sure to define any parameters, and use a significance level of α = 3%. Do not use any R functions that are designed to perform hypothesis tests.

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Drumpf Country Club has the only golf course in Distopia. The (annual) demand to play a round of golf on its course is given by D (p) = 12000-10p where D (p) is the number of rounds demanded each year if Drumpf charged a uniform price of p per round. The marginal cost of having someone play a round of golf is zero and the annual (fixed) cost of maintaining the golf course is 1000000. Drumpf undertakes some market research and finds that its market demand comes from the aggregation of the demands of 100 golfers who each have an identical demand for rounds of golf given by xh (p) = 120- p/10 where xh (p) is the number of rounds of golf demanded by golfer h, h = 1; : : : ; 100. For the rest of this question assume that instead of all golfers being identical, there are two types of golfers, ìtragicsîand ìdilettantesî. There are fifty of each type and the demand of a tragic is given by xT (p) = 160 -p/10 while that of a dilettante is given by xD (p) = 80 -p/10. Unfortunately for Drumpf, it is not possible to tell beforehand whether a golfer is of either type. (c) (5 points) Explain to Drumpf, what is the most it can set for its (fixed) membership fee if it wishes to attract both types of golfers.

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INSTANT ANSWER

Drumpf Country Club has the only golf course in Distopia. The (annual) demand to play a round of golf on its course is given by D (p) = 12000-10p where D (p) is the number of rounds demanded each year if Drumpf charged a uniform price of p per round. The marginal cost of having someone play a round of golf is zero and the annual (fixed) cost of maintaining the golf course is 1000000. Drumpf undertakes some market research and finds that its market demand comes from the aggregation of the demands of 100 golfers who each have an identical demand for rounds of golf given by xh (p) = 120- p/10 where xh (p) is the number of rounds of golf demanded by golfer h, h = 1; : : : ; 100. (b) (10 points) Imagine you have been hired by Drumpf as a consultant to help it maximize profit. In light of the information above, explain how by not charging golfers for each individual round of golf they play but rather charging them a golf club membership fee that allows the member to play an unlimited number of rounds of golf can increase Drumpfís profit. Calculate the optimal membership fee to charge and carefully explain what impact this would have on the number of rounds of golf demanded and Drumpfís profit.

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INSTANT ANSWER

Drumpf Country Club has the only golf course in Distopia. The (annual) demand to play a round of golf on its course is given by D (p) = 12000-10p where D (p) is the number of rounds demanded each year if Drumpf charged a uniform price of p per round. The marginal cost of having someone play a round of golf is zero and the annual (fixed) cost of maintaining the golf course is 1000000. Drumpf undertakes some market research and finds that its market demand comes from the aggregation of the demands of 100 golfers who each have an identical demand for rounds of golf given by xh (p) = 120- p/10 where xh (p) is the number of rounds of golf demanded by golfer h, h = 1; : : : ; 100. (b) (10 points) Imagine you have been hired by Drumpf as a consultant to help it maximize profit. In light of the information above, explain how by not charging golfers for each individual round of golf they play but rather charging them a golf club membership fee that allows the member to play an unlimited number of rounds of golf can increase Drumpfís profit. Calculate the optimal membership fee to charge and carefully explain what impact this would have on the number of rounds of golf demanded and Drumpfís profit.

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INSTANT ANSWER

Drumpf Country Club has the only golf course in Distopia. The (annual) demand to play a round of golf on its course is given by D (p) = 12000-10p where D (p) is the number of rounds demanded each year if Drumpf charged a uniform price of p per round. The marginal cost of having someone play a round of golf is zero and the annual (fixed) cost of maintaining the golf course is 1000000. Drumpf undertakes some market research and finds that its market demand comes from the aggregation of the demands of 100 golfers who each have an identical demand for rounds of golf given by xh (p) = 120- p/10 where xh (p) is the number of rounds of golf demanded by golfer h, h = 1; : : : ; 100. (b) (10 points) Imagine you have been hired by Drumpf as a consultant to help it maximize profit. In light of the information above, explain how by not charging golfers for each individual round of golf they play but rather charging them a golf club membership fee that allows the member to play an unlimited number of rounds of golf can increase Drumpfís profit. Calculate the optimal membership fee to charge and carefully explain what impact this would have on the number of rounds of golf demanded and Drumpfís profit.

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INSTANT ANSWER

Not all of these problems are directly related to Householder reflectors and Householder QR, but the concepts are important in numerical linear algebra. A block diagonal matrix A ∈ Cm×m is any matrix whose nonzero patterning can be written as a collection of square matrices along a diagonal; for example A =  B 0 0 C  (1) where B ∈ Ck×k and C ∈ Cm−k×m−k. The “0” terms are flexibly defined, representing rectangular matrices of zeros of appropriate shape to pad out A. (A diagonal matrix is a special case of a block diagonal matrix where there are m 1-by-1 blocks.) For this class of matrices, much information about A is directly related to information about B and C. For example, the determinant of A is provably what it might look like if we pretended this was a

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INSTANT ANSWER

Drumpf Country Club has the only golf course in Distopia. The (annual) demand to play a round of golf on its course is given by D (p) = 12000-10p where D (p) is the number of rounds demanded each year if Drumpf charged a uniform price of p per round. The marginal cost of having someone play a round of golf is zero and the annual (fixed) cost of maintaining the golf course is 1000000. Drumpf undertakes some market research and finds that its market demand comes from the aggregation of the demands of 100 golfers who each have an identical demand for rounds of golf given by xh (p) = 120- p/10 where xh (p) is the number of rounds of golf demanded by golfer h, h = 1; : : : ; 100. For the rest of this question assume that instead of all golfers being identical, there are two types of golfers, ìtragicsîand ìdilettantesî. There are fifty of each type and the demand of a tragic is given by xT (p) = 160- p/10, while that of a dilettante is given by xD (p) = 80 -p/10 Unfortunately for Drumpf, it is not possible to tell beforehand whether a golfer is of either type. (c) (5 points) Explain to Drumpf, what is the most it can set for its (fixed) membership fee if it wishes to attract both types of golfers.

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INSTANT ANSWER

[3.18] A function \( f: R^{n} \rightarrow R \) is called a gauge function if it satisfies the following equality: \[ f(\lambda \mathbf{x})=\lambda f(\mathbf{x}) \quad \text { for all } \mathbf{x} \in R^{n} \text { and all } \lambda \geq 0 . \] Further, a gauge function is said to be subadditive if it satisfies the following inequality: \[ f(\mathbf{x})+f(\mathbf{y}) \geq f(\mathbf{x}+\mathbf{y}) \quad \text { for all } \mathbf{x}, \mathbf{y} \in R^{n} . \] Prove that subadditivity is equivalent to convexity of gauge functions.

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[3.8] Let \( f_{1}, f_{2}, \ldots, f_{k}: \quad R^{n} \rightarrow R \) be convex functions. Consider the function \( f \) defined by \( f(\mathbf{x})=\sum_{j=1}^{k} \alpha_{j} f_{j}(\mathbf{x}) \), where \( \alpha_{j}>0 \) for \( j=1,2, \ldots, k \). Show that \( f \) is convex. State and prove a similar result for concave functions.

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[3.6] Let \( S \) be a nonempty convex set in \( R^{n} \), and let \( f: S \rightarrow R \). Show that \( f \) is convex if and only if for any integer \( k \geq 2 \), the following holds true: \( \mathbf{x}_{1}, \ldots, \mathbf{x}_{k} \in S \) implies that \( f\left(\sum_{j=1}^{k} \lambda_{j} \mathbf{x}_{j}\right) \leq \sum_{j=1}^{k} \lambda_{j} f\left(\mathbf{x}_{j}\right) \), where \( \sum_{j=1}^{k} \lambda_{j}=1, \lambda_{j} \geq 0 \) for \( j=1, \ldots \), \( k \)

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