\(y=mx+b\)

\(x=my+b\)

\(my=x-b\)

\(y=\frac{x}{m}-\frac{b}{m}\)

Slope: \(\frac{1}{m}\)

y-intercept: \(\frac{-b}{m}\)

\(x=my+b\)

\(my=x-b\)

\(y=\frac{x}{m}-\frac{b}{m}\)

Slope: \(\frac{1}{m}\)

y-intercept: \(\frac{-b}{m}\)

Question

asked 2021-05-16

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.

A. Let y=f(x) be the equation of C. Find f(x).

B. Find the slope at P of the tangent to C.

C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?

D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.

E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.

Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

A. Let y=f(x) be the equation of C. Find f(x).

B. Find the slope at P of the tangent to C.

C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?

D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.

E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.

Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

asked 2021-03-20

The graph of y = f(x) contains the point (0,2), \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}\), and f(x) is greater than 0 for all x, then f(x)=

A) \(\displaystyle{3}+{e}^{{-{x}^{{2}}}}\)

B) \(\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}\)

C) \(\displaystyle{1}+{e}^{{-{x}}}\)

D) \(\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}\)

E) \(\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}\)

A) \(\displaystyle{3}+{e}^{{-{x}^{{2}}}}\)

B) \(\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}\)

C) \(\displaystyle{1}+{e}^{{-{x}}}\)

D) \(\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}\)

E) \(\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}\)

asked 2021-05-14

For the equation (-1,2), \(y= \frac{1}{2}x - 3\), write an equation in slope intercept form for the line that passes through the given point and is parallel to the graph of the given equation.

asked 2021-05-05

The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).

(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.

(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).

(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.

(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?

(e) What is the moment generating function for X?

(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.

(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).

(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.

(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?

(e) What is the moment generating function for X?

asked 2021-05-11

Find the absolute maximum value and the absolute minimum value, if any, of the function.

\(f(x)=8x-\frac{9}{x}\)

\(f(x)=8x-\frac{9}{x}\)

asked 2021-05-20

Assume that a ball of charged particles has a uniformly distributednegative charge density except for a narrow radial tunnel throughits center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton any where along the tunnel or outside the ball. Let \(\displaystyle{F}_{{R}}\) be the magnitude of the electrostatic force on the proton when it islocated at the ball's surface, at radius R. As a multiple ofR, how far from the surface is there a point where the forcemagnitude is 0.44FR if we move the proton(a) away from the ball and (b) into the tunnel?

asked 2021-06-05

Find the absolute maximum value and the absolute minimum value, if any, of the function.

\(g(x)=-x^{2}+2x+6\)

\(g(x)=-x^{2}+2x+6\)

asked 2021-05-11

Consider the polynomial function \(p(x)=(3x^{2}-5x-2)(x-5)(x^{2}-4)\).

A) what is the degree of the polynomial? What is the y intercept of the function?

B)What are the zeros and their multiplicities?

C) what is the leading term of the polynomial and what power function has a graph most similiar to the graph of p?

A) what is the degree of the polynomial? What is the y intercept of the function?

B)What are the zeros and their multiplicities?

C) what is the leading term of the polynomial and what power function has a graph most similiar to the graph of p?

asked 2021-02-25

We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:

a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.

b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.

c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.

d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.

Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.

a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.

b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.

c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.

d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.

Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.

asked 2021-05-12

Find the absolute maximum and absolute minimum values of f on the given interval and state where those values occur: \(f(x)=x^{3}-3x^{2}-9x+25, [-5,10]\)