Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = t2 i + 2t j + ln t k
> 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 at the given point. -k, (1, 1, 1)
> 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) = 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.
> A particle moves with position function r(t) = t ln t i + t j + e-t k. Find the velocity, speed, and acceleration of the particle.
> Find an equation of the osculating circle of the curve y = x4 - x2 at the origin. Graph both the curve and its osculating circle.
> Find the curvature of the curve y = x4 at the point (1, 1).
> Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4).
> Reparametrize the curve r(t) = et i + et sin t j + et cos t k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t.
> How do you find the length of a space curve given by a vector function r(t)?
> How do you find the tangent vector to a smooth curve at a point? How do you find the tangent line? The unit tangent vector?
> (a). Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b). Use a computer to graph the path of the particle. a(t) = t i + et j + e-t k, v (0) = k, r (0) = j + k
> Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable. z= 19
> Prove that if f is a function of two variables that is differentiable at (a, b), then f is continuous at (a, b). Hint: Show that lim (Ax. Ay)-(0, 0) f(a + Ax, b + Ay) = f(a, b)
> Show that the function is differentiable by finding values of «1 and «2 that satisfy Definition 7. f (x, y) = xy - 5y2
> Show that the function is differentiable by finding values of «1 and «2 that satisfy Definition 7. f (x, y) = x2 + y2
> Suppose you need to know an equation of the tangent plane to a surface S at the point P (2, 1, 3). You don’t have an equation for S but you know that the curves both lie on S. Find an equation of the tangent plane at P. г.() — (2 +
> A model for the surface area of a human body is given by S = 0.1091w0.425h0.725, where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. If the errors in measurement of w and h are at most 2%, use differentials t
> If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2, R3, then If the resistances are measured in ohms as R1 = 25 V, R2 = 40 V, and R3 = 50 V, with a possible error of 0.5% in each case, estimate the maximum e
> The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV = 8.31T, where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the volume incr
> The tension T in the string of the yo-yo in the figure is where m is the mass of the yo-yo and t is acceleration due to gravity. Use differentials to estimate the change in the tension if R is increased from 3 cm to 3.1 cm and r is increased from 0.7 c
> The length and width of a rectangle are measured as 30 cm and 24 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.
> (a). Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b). Use a computer to graph the path of the particle. A(t) = 2t i + sin t j + cos 2t k, v (0) = i, r (0) = j
> 1. Expression 1 gives the fractional change in speed that results from a change x in power and a change y in drag. Show that this reduces to the function Given the context, what is the domain of f ? 2. Suppose that the possible changes in power x and d
> Different parametrizations of the same curve result in identical tangent vectors at a given point on the curve.
> If |r(t) | = 1 for all t, then |r'(t) | is a constant.
> Show that the curve with vector equation lies in a plane and find an equation of the plane. r(t) = (aıt? + bit + cı, azt² + bat + C2, ast² + bạt + cs)
> A cable has radius r and length L and is wound around a spool with radius R without over- lapping. What is the shortest length along the spool that is covered by the cable?
> A ball rolls off a table with a speed of 2 ft/s. The table is 3.5 ft high. (a). Determine the point at which the ball hits the floor and find its speed at the instant of impact. (b). Find the angle between the path of the ball and the vertical line dr
> Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = sin t i + 2 cos t j + 6t k, v (0) = 2k, r (0) = j - 4 k
> Find a vector function that represents the curve of intersection of the cylinder x2 + y2 = 16 and the plane x + z = 5.
> A projectile is launched with an initial speed of 40 m/s from the floor of a tunnel whose height is 30 m. What angle of elevation should be used to achieve the maximum possible horizontal range of the projectile? What is the maximum range?
> An athlete throws a shot at an angle of 450 to the horizontal at an initial speed of 43 ft/s. It leaves his hand 7 ft above the ground. (a). Where is the shot 2 seconds later? (b). How high does the shot go? (c). Where does the shot land?
> The figure shows the curve C traced by a particle with position vector r(t) at time t. (a) Draw a vector that represents the average velocity of the particle over the time interval 3 < t < 3.2. (b) Write an expression for the velocity v (3). (c) Write an
> Let C be the curve with equations x = 2 - t3, y = 2t - 1, z − ln t. Find (a). the point where C intersects the xz-plane, (b). parametric equations of the tangent line at (1, 1, 0), and (c). an equation of the normal plane to C at (1, 1, 0).
> Find parametric equations for the tangent line to the curve x = 2 sin t, y = 2 sin 2t , z = 2 sin 3t at the point (1, 3 , 2). Graph the curve and the tangent line on a common screen.
