Question: Find the tangential and normal components of

> Find the first partial derivatives of the function. f (x, t) = t2e-x

> Find the first partial derivatives of the function. f (x, y) = x2y - 3y4

> Find the first partial derivatives of the function. f (x, y) = x4 + 5xy3

> The wind-chill index is modeled by the function W = 13.12 + 0.6215T - 11.37v0.16 + 0.3965Tv 0.16 where T is the temperature (°C) and v is the wind speed (km/h). When T − 215°C and v − 30 km/h, by how much would you expect the apparent temperature W to

> For the ideal gas of Exercise 88, show that Exercise 88: The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Show that ,aP av = mR aT ƏT aP av əT -1 av aT əP

> The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Show that aP av əT -1 av aT əP

> A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated

> Find the limit, if it exists, or show that the limit does not exist. x²y + xy² lim x² – y² (x, y)(2, -1) x – y?

> Find the limit, if it exists, or show that the limit does not exist. (x²y³ – 4y²) (х, у)— (3, 2)

> At the beginning of this section we considered the function and guessed on the basis of numerical evidence that f (x, y) → 1 as (x, y) → (0, 0). Use polar coordinates to confirm the value of the limit. Then graph the f

> Determine the set of points at which the function is continuous. F(x, y) 1+ x² + y? 1– x² – y2

> Determine the set of points at which the function is continuous. F(x, y) = cos /1 + x – y

> Find h (x, y) = g (f (x, y)) and the set of points at which h is continuous. g(t) = t² + JT, ƒ(x, y) = 2x + 3y – 6

> A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 508 above the horizontal. Is it a home run? (In other words, does the ba

> Use a computer graph of the function to explain why the limit does not exist. lim (x, y) (0, 0) 2.x? + 3xy + 4y² 3x? + 5y?

> Find the limit, if it exists, or show that the limit does not exist. x?y?z? lim (3, x 2(0, 0, 0) x? + y? + z?

> Find the limit, if it exists, or show that the limit does not exist. ху + yz lim (x, y, z-(0,0,0) x² +: y? + z?

> Find the limit, if it exists, or show that the limit does not exist. lim (x, y, 2)-(m, 0, 1/3) e tan(xz)

> Find the limit, if it exists, or show that the limit does not exist. x² + y? lim (x, y)-(0, 0) x² + y² + I – 1

> Find the limit, if it exists, or show that the limit does not exist. ху lim (x, y)-(0, 0) x* + y4

> Find the limit, if it exists, or show that the limit does not exist. xy cos y lim + y* .2 (x, y) (0, 0) x

> Describe the level surfaces of the function. f (x, y, z) = y2 + z2

> Describe the level surfaces of the function. f (x, y, z) = x2 + 3y2 + 5z2

> Describe the level surfaces of the function. f (x, y, z) = x + 3y + 5z

> A rifle is fired with angle of elevation 36°. What is the muzzle speed if the maximum height of the bullet is 1600 ft?

> Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices. B D E II III IV VI yA (OKO C O O O O).O O O) O O KO C O OD O x- y 1+ x² + y²

> Draw a contour map of the function showing several level curves. f(x, y) = xy

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f.

> Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices. z = sin (x – y) B D E II III IV VI yA (OKO C O O O O).O O O

> Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices. z = ex cos y B D E II III IV VI yA (OKO C O O O O).O O O) O O KO C O OD O

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f. y.

> Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the

> Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the

> Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f. 14, * 13 12

> A company makes three sizes of cardboard boxes: small, medium, and large. It costs $2.50 to make a small box, $4.00 for a medium box, and $4.50 for a large box. Fixed costs are $8000. (a). Express the cost of making x small boxes, y medium boxes, and z l

> A thin metal plate, located in the xy-plane, has temperature T (x, y) at the point (x, y). Sketch some level curves (isothermals) if the temperature function is given by 100 T(x, y) = 1+ x² + 2y²

> Make a rough sketch of a contour map for the function whose graph is shown.

> Locate the points A and B on the map of Lonesome Mountain (Figure 12). How would you describe the terrain near A? Near B? Figure 12: 90 90 180 40 60 60- -60 70 50 60 80 70 60 70 80 70 30- -30 -100 80 0- -- 70 80 30- 60 -30 50 40 60- -60 30 20 10 90

> Draw a contour map of the function showing several level curves. f (x, y) = y/ (x2 + y2)

> Draw a contour map of the function showing several level curves. f(x, y) = x² + y?

> Draw a contour map of the function showing several level curves. f (x, y) = y - arctan x

> Draw a contour map of the function showing several level curves. f(x, y) = ye²

> Draw a contour map of the function showing several level curves. f(x, y) = In(x² + 4y³)

> Draw a contour map of the function showing several level curves. f(x, y) = VF + y

> A ball is thrown at an angle of 45° to the ground. If the ball lands 90 m away, what was the initial speed of the ball?

> A rocket burning its onboard fuel while moving through space has velocity v(t) and mass m(t) at time t. If the exhaust gases escape with velocity ve relative to the rocket, it can be deduced from Newton’s Second Law of Motion that (b

> Draw a contour map of the function showing several level curves. f(x, y) = x² – y?

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f. 5 3 2 15 3 2

> Find the tangential and normal components of the acceleration vector. r(t) = t i + 2et j + e2t k

> Find the tangential and normal components of the acceleration vector. r(t) = cos t i + sin t j + t k

> Find the tangential and normal components of the acceleration vector. r(t) = 2t2 i + ( 2 3 t3 - 2t) j

> Find the tangential and normal components of the acceleration vector. r(t) = (t2 + 1) i + t3 j, t > 0

> Rework Exercise 23 if the projectile is fired from a position 100 m above the ground. Exercise 23: A projectile is fired with an initial speed of 200 m/s and angle of elevation 60°. Find (a) the range of the projectile, (b) the maximum height reached,

> Determine the set of points at which the function is continuous. f(x, y, z) = Vy – x² In z

> Find the limit, if it exists, or show that the limit does not exist. x³ - y3 lim (x, y)-0, 0) x? + xy + y? .2 2

> Find the limit, if it exists, or show that the limit does not exist. ху — у lim (x, 3)(1, 0) (x – 1)² + y?

> Determine the set of points at which the function is continuous. G(x, y) = Vx + /T- x² – y²

> Sketch the graph of the function. f (x, y) = cos y

> Sketch the graph of the function. f(x, y) = V4 – 4x² – y²

> Sketch the graph of the function. f(x, y) = /4x2 + y?

> Sketch the graph of the function. f (x, y) = x2 + 4y2 + 1

> Sketch the graph of the function. f (x, y) = 2 - x2 - y2

> Sketch the graph of the function. f (x, y) = sin x

> A projectile is fired with an initial speed of 200 m/s and angle of elevation 60°. Find (a) the range of the projectile, (b) the maximum height reached, and (c) the speed at impact.

> A projectile is fired from a tank with initial speed 400 m/s. Find two angles of elevation that can be used to hit a target 3000 m away.

> Sketch the graph of the function. f (x, y) = 10 - 4x - 5y

> Find the limit, if it exists, or show that the limit does not exist. y? sin?x lim (x, y)(0, 0) x* + y

> Sketch the graph of the function. f (x, y) = y

> Find and sketch the domain of the function. f (x, y, z) = ln (16 - 4x2 - 4y2 - z2)

> Find the limit, if it exists, or show that the limit does not exist. lim (x, y, z)-(0, 0, 0) xy + yz? + xz2 x² + y² + : 4 z

> What force is required so that a particle of mass m has the position function r(t) = t3 i + t2 j + t3 k?

> Find and sketch the domain of the function. f(x, y) = Vy – x2 y - 1- x?

> A model for the surface area of a human body is given by the function where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. (a). Find f (160, 70) and interpret it. (b). What is your own surface area? S= f

> Find and sketch the domain of the function. х — у g(x, y) x + y

> Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.

> Find and sketch the domain of the function. f(x, y) = Vx? + y² – 4

> Find and sketch the domain of the function. f (x, y) = ln (9 - x2 - 9y2)

> Find and sketch the domain of the function. f(x, y) = Vx – 3y

> Find and sketch the domain of the function. f(x, y) = Vx – 2 + Jy – 1

> Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = t2 i + 2t j + ln t k

> Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = 2 t i + et j + e-t k

> The binormal vector is B(t) = N(t) × T(t).

> If T(t) is the unit tangent vector of a smooth curve, then the curvature is k = |dT/dt |.

> If r(t) is a differentiable vector function, then d |r()| = |r'(t)| dt

> A force with magnitude 20 N acts directly upward from the xy-plane on an object with mass 4 kg. The object starts at the origin with initial velocity v (0) = i - j. Find its position function and its speed at time t.

> If u(t) and v(t) are differentiable vector functions, then d [u(t) x v()] = u'(1) × v'(t) dt

> The derivative of a vector function is obtained by differentiating each component function.

> The osculating circle of a curve C at a point has the same tangent vector, normal vector, and curvature as C at that point.

> If |r(t) | = 1 for all t, then r'(t) is orthogonal to r(t) for all t.

> If k(t) = 0 for all t, the curve is a straight line.

> The curve with vector equation r(t) − t3 i + 2t3 j + 3t3 k is a line.

> Find the curvature of the curve with parametric equations [ sin(}w0³) do y = cos(0) de %3D

> Find and sketch the domain of the function. f (x, y) = sin-1(x + y)

> The figure shows the path of a particle that moves with position vector r(t) at time t. (a). Draw a vector that represents the average velocity of the particle over the time interval 2 (b). Draw a vector that represents the average velocity over the ti

> The helix r1(t) = cos t i + sin t j + t k intersects the curve r2(t) = s1 + td i + t2 j + t3 k at the point (1, 0, 0). Find the angle of intersection of these curves.

> Use Simpson’s Rule with n = 6 to estimate the length of the arc of the curve with equations x = t2, y = t3, z = t4, 0 < t < 3.

> Find the tangential and normal components of the acceleration vector of a particle with position function r(t) = t i + 2t j + t2 k

> A particle starts at the origin with initial velocity i - j + 3k. Its acceleration is a(t) = 6t i + 12t2 j - 6t k. Find its position function.

> Find the velocity, speed, and acceleration of a particle moving with position function r(t) = (2t2 – 3) i + 2t j. Sketch the path of the particle and draw the position, velocity, and acceleration vectors for t = 1.