Basic Math question and answers for August 12, 2023
- Q Let the natural number n have the decimal numeral 123,461,46d,where d is the units digit. Use divisibility tests to give all ofthe choices of d by which n is divisible....
- Q The velocity function of a particle moving along a line is givenby the equation v(t) = t2 - 2t -3. The particle hasinitial position s(0) = 4.a. Find the displacement...
- Q Let F(x)=f(f(x)) and G(x)=(F(x))^2 . You also know thatf(6)=14,f(14)=3,f′(14)=4,f′(6)=3.Find F′(6)= and G′(6)= .
- Q Determine the tangent line approximation for ?(?) = 2sin(? ? )at the point (0, 2???1)
- Q Take the Laplace transform of the following initial value andsolve for X(s)=L{x(t)}X(s)=L{x(t)}:x′′+16x={sin(πt),0}0≤t<11≤tx(0)=0x′(0)=0.a) X(s)=  Now find the inverse transform to findb) x(t)=Use u(t−a) for the Heaviside function shifted a unitshorizontaly.
- Q Farmer S. Unkist has a fruit grove consisting of lemons,bananas, and watermelons along a straight moat. To prevent thievesfrom stealing his fruit while at the same time make sure the...
- Q A cup of coffee is made with boiling water at 100 and stands ina room where the temperature is 25. The change in temperature, H indegrees , with respect to...
- Q Suppose ?(?) represents the number of bacteria in a dish t hoursafter the start of a lab experiment. a) Interpret the meaning of?(20) = 100100 is the average rate of...
- Q Solve the initial–value problem by using the Laplacetransform.y′′ +2y′ +10y = δ(t−π), y(0) = 0, y′(0) = 4
- Q Solve the IVP: y’’’ – y ’= 2sinxwhere: y(0)=0, y’(0)=0, yâ€(0)=1Use an annihilator method, please.
- Q 3. Let g(θ) = 2 cos(θ) + sin(2θ) . Find the absolute maximum andminimum values of g on the interval [0, π/2]
- Q A farmer has produced 1000 apples and wants to sell them. He cansell the apples at two different markets:- At market 1, if the farmer sells x apples, he can...
- Q Please show work.1. Find the derivatives dy/dx and d2y/dx and evaluate them at t= 2.x = t2, y = tln t2. Find the arc length of the curve on the...
- Q For the given function determine the following: f (x) = (sin x +cos x) 2 ; [−π,π] a) Find the intervals where f(x) is increasing,and decreasing b) Find the intervals...
- Q he organizers of a Reindeer Exhibit have asked you to buildtemporary yards for two reindeer; each yard will be surrounded bycandy-cane fencing on all sides. The geometrically-minded reindeeronly like to...
- Q 1. An astronaut with a spaceship moves from left to rightfollowing the curve y = x^2. When he turns off the engine, thespacecraft will move along the tangent line at...
- Q Let R(x) = ( 2x^2 − 8x + 6 ) / (x^2 + 6x + 8)a) Find all x such that R(x) is undefined. Find all x such thatR(x) =...
- Q Plot the limaçon defined by polar equation r=1+2sin(Ѳ) and findthe area inside the limaçon but outside of the inner loop.
- Q 2. Evaluate the following limits.(a) lim x→1+ln(x) /ln(x − 1)(b) limx→2πx sin(x) + x ^2 − 4π^2/x − 2π(c) limx→0sin^2 (3x)/x^2
- Q The function f(x)= x^−5 has a Taylor series at a=1 . Find thefirst 4 nonzero terms in the series, that is write down the Taylorpolynomial with 4 nonzero terms.
- Q use a Taylor series to get the diverivative of f(x)=arctanx^3and check for the interval of convergence. Is the interval ofconvergence for f' the same as the interval for f or...
- Q 1. (a) Find the directional derivative of f at (2,1) in thedirection from (2,1) to (6,-2). Show your work clearly.(b) Find a unit vector u such that the directional derivative...
- Q 1. write a sixteen line about Taylor's theorem2. write a haiku about exponential growth
- Q Find the Maclaurin series of the functionf(x)=(3x2)e^−4x(f(x)=∑n=0∞cnxn)
- Q Verify the Divergence Theorem for the vector field F(x, y, z) =< y, x , z^2 > on the region E bounded by the planes y + z =2, z...
- Q  Estimate ∫^−3 −5 ?^2+5? ?? using midpoints for ?=4n=4 approximating rectangles.∫^−3 −5 ?^2+5? ?? is approximatelyEstimate ∫^3 2 2/? ?? using right endpoints for ?=3 approximating rectangles.∫^3 2 2/? ??...
- Q Find an approximation of the area of the region R underthe graph of the function f on the interval [1, 3]. Usen = 4 subintervals. Choose the representative points to...
- Q 1. rewrite as a single logarithm: 4ln(x-2x)-3lnx+ln72. Solve using common base: 43x-1=1653. Solve using logarithm: 94x-3=8454. 6log4(7x)=-18
- Q 1. Use the derivative function, f'(x)f′(x), to determine wherethe functionf(x)=−2x^2+14x−8is increasing.2.Use the derivative function f'(x)f′(x) to determine where thefunction f(x)=2x^3−27x^2+108x+13 is increasing.  3.Use the derivative function f'(x)f′(x) to determine where thefunction...
- Q Let f(x, y) = − cos(x + y2) and let a be the point a = ( π/2,0).(a) Find the direction in which fincreases most quickly at the pointa.(b) Find...
- Q . Let two circles C1 and C2 intersect at X and Y. Prove that apoint P is on the line XY if and only if the power of P withrespect...
- Q 1. Which of the following is the linear approximation ofthe function f ( x ) = 2e^sin (7x) at x = 0?Group of answer choicesy=cosâ¡(7)x+2y=7x+2y=14x+2y=2x+7y=e^7x+142. Recall that Rolle's Theorem begins,...
- Q Matlab code please6. Find the velocity, acceleration, and speed of a particle withthe given position function. (a) r(t) = e t cos(t)i+e t sin(t)j+tetk t = 0 (b) r(t) =...
- Q Derivatives of Exponential Functions(Scenario)Consider the following problem: Supposethat F(x) computes the rabbit population on a game reservethat doubles every 6 months. Suppose there were 200 rabbitsinitially.1. Write a mathematical expression...
- Q 1. Find the intervals on which the graph of fis concave​ upward, the intervals on which the graph of f isconcave​ downward, and the inflection points. f(x)= -x^4 + 12x^3...
- Q If f(x)=2x^2−5x+3, findf'(−4).     Use this to find the equation of the tangent line to the parabolay=2x^2−5x+3 at the point (−4,55). The equation of this tangent linecan be written in the form...
- Q Find the absolute extrema if they exists, as well as all valuesof x, where they occur, for the function f(x)=1/3x^3+1/2x^2-12x+5on domain [6,6]
- Q Evaluate the line integral∫CF⋅dr∫CF⋅dr,whereF(x,y,z)=sin(x)i+cos(y)j+4xzkF(x,y,z)=sin(x)i+cos(y)j+4xzkand C is given by the vector functionr(t)=t3i−t2j+tkr(t)=t3i−t2j+tk, 0≤t≤10≤t≤1.
- Q The following table provides the number of men infected with HIVin San Francisco from 1982 to 1990. Year Quantity Delaware infected1982 80 1983 300 1984 700 1985 1500 1986 2500...
- Q 1. A spherical balloon is being inflated so that itsvolume increases at a rate of 1200cm^3 per minute. How fast is itsradius increasing at the moment when the radius is...
- Q Find the value of Δz for the function f(x,y)=2x2+xy2+y at point(1,1). Use Δx=0.02 and Δy=0.02. Round to three decimal places.
- Q a) The number of voices I hear at 7am is 12000 at 9am the countis 36000, if the number of voices grows exponentially, find thetime it takes to reach 81000...
- Q Suppose that the mass in a mass-spring-dashpot system with m =10, the damping constant c = 9, and the spring constant k = 2 isset in motion with x(0) =...
- Q In the first 5 examples find the derivative ofYY=epower2xY=epower-2x +xpower2Y=epowerxpower1/2Y= epower-3/xY=epower4xdivided by xpower2Integrate the followingInt x epower5xpower2 dxInt epower-2x between 1 and 0Int.epower x1/2 divided by xpower1/2
- Q find the curvature and torsion of beta(t) = (e^tcost, e^tsint,e^t)
- Q In general:1) What are gradient vectors?a. How are they formed?b. Where are they located?c. How are they related to level curves?d. What direction do they point?e. Why are they important?
- Q There is a line through the origin that divides the regionbounded by the parabola y=8x−7x^2 and the x-axis into tworegions with equal area. What is the slope of that line?
- Q A rectangular box with a square base has a volume of 4 cubicfeet. The material for the bottom of the box costs $3 per squarefoot, the top costs $2 per...
- Q A rectangular box with a square base has a volume of 4 cubicfeet. The material for the bottom of the box costs $3 per squarefoot, the top costs $2 per...
- Q Where is the function increasing or decreasing and what are therelative mins and max?g(t) = 3t^4-16t^3+24t^2, Domain (-infinity, +infinity)k(x) = 2x/5 - (x-1)^2/5, Domain [0, infinity)g(x) = x^3/x^2-3g(x) = x^2-4√...
- Q -Use implicit differentiation to find dy/dx wherex2y2 + x cos(y) = 6-Use logarithmic differentiation to find the derivative of o(x)= (cos(x))ln(x)-Find the first and second derivatives of p(x) = 6ln(x2+3x)-A...
- Q Calculate the directional derivative off(x,y,z)=x(y^2)+y((1-z)^(1/2)) at the point P(1,−2,0) in thedirection of the vector v = 5i+2j−k. (a) Calculate the directionalderivative of f at the point P in the direction...
- Q The position of a particle moving along the x-axis is given byx(t) = t^3 + 9t^2 − 21t with t is in [0, 2]. (a) Find the velocityand acceleration of...
- Q Find f1)fâ€(x)=-2+12x-12x(Square), f’(0)=12F(x)=?2)Find a function f such that f’(x)=5x cube and the line5x+y=0 is the tangent to the graph of fF(x)=?3)A particle is moving with the given data. Find the...
- Q Fit a quadratic function of the form ?(?)=?0+?1?+?2?2 f ( t ) =c 0 + c 1 t + c 2 t 2 to the data points (0,−1) ( 0...
- Q When a cold drink is taken from a refrigerator, its temperatureis 5C. After 25 minutes in a 20C room, its temperature hasincreased to 10C.a. what is the temperature of the...
- Q Determine if the following series converges or diverges andexplain why.1) 1 - (4/6) + (4/6)^2 - (4/6)^3 + ...2) 1 - (6/4) + (6/4)^2 - (6/4)^3 + ...
- Q The temperature T in a metal ball is inversely proportional tothe distance from the center of the ball, which we take to be theorigin. The temperature at the point (1,...
- Q Find the exact area of the surface obtained by rotating thecurve about the x-axis.y =1 + ex,    0 ≤ x ≤ 6
- Q Evaluate the line integral ∫_C F⋅dr, whereF(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vectorfunction r(t)=t^6i−t^5j+t^4k , 0≤t≤1.
- Q Â part 1)Gravel is being dumped from a conveyor belt at a rate of 40 ft3/min. It forms a pile in the shape of a right circular cone whose base diameter...
- Q Aki’s bicycle designs has determined that when x hundredbicycles are built, the average cost per bicycle is given byC(x)=0.2x^2-0.1x+5.857, where C(x) is in hundreds of dollars.How many bicycles should the...
- Q A rock is thrown upward from a bridge that is 31 feet above aroad. The rock reaches its maximum height above the road 0.57seconds after it is thrown and contacts...
- Q Â part 1)A road perpendicular to a highway leads to a farmhouse located 1 mile away. An automobile traveling on the highway passes through this intersection at a speed of 65mph....
- Q Consider the triangle which hasa=4, b=7 and c=8Find the measure of angles ∠A=, ∠B= and ∠C= Give your answer indegrees to at least 3 decimal places.
- Q Use the function f(x) = 2x3 - 6x2. Clearly mark your answer foreach part.Find the x and y intercepts. Give your answer as points.State the intervals of increase and decrease.State...
- Q Take a circle tangent to three lines and located outside of thetriangle defined by these lines. Take the three points of tangencyand prove that the three corresponding Chevians pass through...
- Q Find the area between the curve and the​ x-axis over theindicated interval.y = 100 − x2​;  ​[−10​,10​]The area under the curve is ___​(Simplify your​ answer.)
- Q A bug is crawling at a velocity of f(t) = 1/1 + t meters perhour, where 0 ≤ t ≤ 1 and t is in hours.a) Find both the left-endpoint...
- Q A company operates two plants which manufacture the same itemand whose total cost functions are C1=6.4+0.04(q1)^2 andC2=7.9+0.02(q2)^2 where q1 and q2 are the quantities produced byeach plant. The company is...
- Q  part 1)Find dy/dx by implicit differentiation.3x^6+x^5y−2xy^6=8dy/dx=part 4)A campground owner has 2000 meters of fencing. She wants to enclose a rectangular field bordering a lake, with no fencing needed along the...
- Q Complete a thorough curve analysis of the function ?(?) =?4 − 12?3 +48?2 − 64?.Follow the Algorithm for Curve Sketching as outlined in Section4.5 of your textbook. You are expected...
- Q The Vector Field f(x, y) = (2x + 2y^2)i + (4xy - 6y^2)j hasexactly one potential function f (x, y) that satisfies f(0, 0).Find this potential function , then find...
- Q Compute the directional derivative ofT(x,y,z)=200e^(-x^2-3y^2-9z^2) at the point (2,-1,2) in thedirection toward the point (3,-3,3). What is the maximum rate ofchange of T at the point (2,-1,2)?I have solve this...
- Q onemodel for the spread of a rumor is that the rate of spread isproportional to the product of the fraction y of the population whohave heard the rumor and the...
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