The curves r1(t) = < t, 4, t2-9 > and r2(s) = < 3, s2, 4-2s...

The curves r₁(t) = < t, 4, t²-9 > and r₂(s) = < 3, s², 4-2s >lie on surface of f and intersect at ( 3, 4, 0 ). Find a linear approximation for the # f(3.1, 4.1).

Expert Solution

given r₁(t) = < t, 4, t²-9 > and r₂(s) = < 3, s², 4-2s >

at (3,4,0)
t=3
4-2s=0
=>2s=4
=>s=2

at the point (3,4,0) , correspoindind values of t and s on r₁(t) = < t, 4, t²-9 > and r₂(s) = < 3, s², 4-2s > are t=3,s=2

r₁(t) = < t, 4, t²-9 > and r₂(s) = < 3, s², 4-2s >
=>r'₁(t) = < 1, 0, 2t-0 > and r'₂(s) = < 0, 2s, 0-2 >
=>r'₁(t) = < 1, 0, 2t > and r'₂(s) = < 0, 2s, -2 >
=>r'₁(3) = < 1, 0, 2(3) > and r'₂(2) = < 0, 2(2), -2 >
=>r'₁(3) = < 1, 0, 6 > and r'₂(2) = < 0, 4, -2 >

r'₁(3) = < 1, 0, 6 >
r'₂(2) = < 0, 4, -2 >

(r'₁(3))X(r'₂(2)) =<(0*-2)-(4*6),(0*6)-(1*-2),(1*4)-(0*0)>
=>(r'₁(3))X(r'₂(2)) =<-24,2,4>

equation of tangent plane to the surface at (3,4,0):

<-24,2,4>.<x-3,y-4,z-0>=0
=>-24(x-3)+2(y-4)+4(z-0)=0
=>4(z-0)=24(x-3)-2(y-4)
=>(z-0)=6(x-3)-0.5(y-4)
=>z=6x-18-0.5y+2
=>z=6x-0.5y-16

linear approximation:

f(x,y)=6x-0.5y-16
=>f(3.1, 4.1)≈6(3.1)-0.5(4.1)-16
=>f(3.1, 4.1)≈18.6-2.05-16
=>f(3.1, 4.1)≈0.55

milcah answered 2 years ago

At what point do the curves r1(t) = t, 3 − t, 35 + t2 and...

At what point do the curves r1(t) = t, 3 − t, 35 + t2 and r2(s) = 7 − s, s − 4, s2 intersect? (x, y, z) = Find their angle of intersection, θ, correct to the nearest degree. θ = °

Two spacecraft are following paths in space given by r1=〈sin(t),t,t2〉r1=〈sin⁡(t),t,t2〉 and r2=〈cos(t),1−t,t3〉.r2=〈cos⁡(t),1−t,t3〉. If the temperature for...

Two spacecraft are following paths in space given by r1=〈sin(t),t,t2〉r1=〈sin⁡(t),t,t2〉 and r2=〈cos(t),1−t,t3〉.r2=〈cos⁡(t),1−t,t3〉. If the temperature for the points is given by T(x,y,z)=x2y(6−z),T(x,y,z)=x2y(6−z), use the Chain Rule to determine the rate of change of the difference DD in the temperatures the two spacecraft experience at time t=2.t=2. (Use decimal notation. Give your answer to two decimal places.) dDdt=dDdt=

Find the intersection point (if any) of the lines r1(t)=(17,54,−22)+t(−3,−8,3)r1(t)=(17,54,−22)+t(−3,−8,3) and r2(s)=(−67,−50,37)+s(12,8,−7)r2(s)=(−67,−50,37)+s(12,8,−7). Please show full working/steps...

Find the intersection point (if any) of the lines r1(t)=(17,54,−22)+t(−3,−8,3)r1(t)=(17,54,−22)+t(−3,−8,3) and r2(s)=(−67,−50,37)+s(12,8,−7)r2(s)=(−67,−50,37)+s(12,8,−7). Please show full working/steps to help with learning

Find the inverse Laplace transform of a. F(s) = se-6s/(s-2)(s2+2s+1) b. F(s) = e-12s/s3(s2+4)

M (g) M (kg) Delta t1(s) Delta t2(s) Delta t3(s) Delta t(s) avg. T (s) T2...

M (g) M (kg) Delta t1(s) Delta t2(s) Delta t3(s) Delta t(s) avg. T (s) T2 (s2) 60 0.06 6.53 6.55 6.31 6.46 0.646 0.42 80 0.08 6.63 6.60 6.64 6.62 0.662 0.44 100 0.10 6.99 6.87 7.08 6.98 0.698 0.49 120 0.12 8.12 8.36 8.46 8.31 0.831 0.69 140 0.14 8.35 8.11 8.15 8.20 0.820 0.67 find the slope (4pi^2)/k calculate the intercept ((4pi^2)/3k)m Equate the value of the slope found at point 5 with (4pi^2)/k and solve for...

3. Use Matlab to find the partial fraction expansion of the functions below. a) F(s)=16(s+2)/((s+4)(s2+6s+9)) show...

3. Use Matlab to find the partial fraction expansion of the functions below. a) F(s)=16(s+2)/((s+4)(s2+6s+9)) show your Matlab commands and answers in the space below b) F(s)=(s2+2s+2)/((s+1)2(s+4)2) show your Matlab commands and answers in the space below

Activity 4: • What was the peak voltage at R1 and R2 when you first stimulated...

Activity 4: • What was the peak voltage at R1 and R2 when you first stimulated the neuron using 30mV? • What does TTX do to voltage-gated Na+ channels? • What does lidocaine do to voltage-gated Na+ channels? How does the effect of lidocaine differ from the effect of TTX? • Can someone affected by TTX be treated? • Pain-sensitive neurons (called nociceptors) conduct action potentials from the skin or teeth to sites in the brain involved in pain perception....

A system has the transfer function H(s)=3(s2+7 s)/(s2+8s+4). Draw a Bode plot of the transfer function...

A system has the transfer function H(s)=3(s2+7 s)/(s2+8s+4). Draw a Bode plot of the transfer function and classify it as lowpass, highpass, bandpass, or bandstop. Draw the Direct Form II for this system.

3. Assume an annulus of inner radius r1 and outer radius r2. The inner surface is...

3. Assume an annulus of inner radius r1 and outer radius r2. The inner surface is at T1, the outer surface at T2, T1 > T2. Assume heat transfer between the surfaces by conduction, with a variable conductivity, k = a + bT, develop an expression for the temperature in the material of the annulus.

9) R is the region bounded by the curves ? = x^3 , y=2x+4 , And...

9) R is the region bounded by the curves ? = x^3 , y=2x+4 , And the y-axis. a) Find the area of the region. b) Set up the integral you would use to find the volume of a solid that has R as the base and square cross sections perpendicular to the x-axis.

Question