(a). How do you find the velocity, speed, and acceleration of a particle that moves along a space curve? (b). Write the acceleration in terms of its tangential and normal components
> 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.
> (a). Write formulas for the unit normal and binormal vectors of a smooth space curve r(t). (b). What is the normal plane of a curve at a point? What is the osculating plane? What is the osculating circle?
> State Kepler’s Laws.
> Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 2 i + 2t k, v (0) = 3 i - j, r (0) = j + k
> Find an equation of the tangent plane to the given surface at the specified point. z = x sin (x + y), (-1, 1, 0)
> Find an equation of the tangent plane to the given surface at the specified point. z = x/y2, (-4, 2, -1)
> Find an equation of the tangent plane to the given surface at the specified point. z = ex-y, (2, 2, 1)
> Find an equation of the tangent plane to the given surface at the specified point. z = (x + 2)2 – 2 (y – 1)2 - 5, (2, 3, 3)
> Find an equation of the tangent plane to the given surface at the specified point. z = 2x2 + y2 - 5y, (1, 2, -4)
> (a). The function was graphed in Figure 4. Show that fx (0, 0) and fy (0, 0) both exist but f is not differentiable at s0, 0d. [Hint: Use the result of Exercise 45.] Figure 4: Exercise 45: Prove that if f is a function of two variables that is diffe
> In Exercise 14.1.39 and Example 14.3.3, the body mass index of a person was defined as B (m, h) = m/h2, where m is the mass in kilograms and h is the height in meters. Exercise 14.1.39: The body mass index (BMI) of a person is defined by where m is the
> The wind-chill index is modeled by the function where T is the temperature (in 0C) and v is the wind speed (in km/h). The wind speed is measured as 26 km/h, with a possible error of 62 km/h, and the temperature is measured as -110C, with a possible err
> Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick.
> Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick.
> If z = x2 - xy + 3y2 and (x, y) changes from (3, -1) to (2.96, 20.95), compare the values of ∆z and dz.
> The wind-chill index W is the perceived temperature when the actual temperature is T and the wind speed is v, so we can write W = f (T, v). The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to find a linear approxima
> Use the table in Example 3 to find a linear approximation to the heat index function when the temperature is near 948F and the relative humidity is near 80%. Then estimate the heat index when the temperature is 958F and the relative humidity is 78%.
> The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h = f (v, t) are recorded in feet in the following table. Use the table to find a linear ap
> Given that f is a differentiable function with f (2, 5) = 6, fx (2, 5) = 1, and fy (2, 5) = -1, use a linear approximation to estimate f (2.2, 4.9).
> Verify the linear approximation at (0, 0). y - 1 -х+у—1 х+1
> Explain why the function is differentiable at the given point. Then find the linearization L (x, y) of the function at that point. f (x, y) = y + sin (x/y), (0, 3)
> Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point. f (x, y) = x2ey, (1, 0)
> Draw the graph of f and its tangent plane at the given point. (Use your computer algebra system both to compute the partial derivatives and to graph the surface and its tangent plane.) Then zoom in until the surface and the tangent plane become indisting
> Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = et (cos t i + sin t j + t k)
> Draw the graph of f and its tangent plane at the given point. (Use your computer algebra system both to compute the partial derivatives and to graph the surface and its tangent plane.) Then zoom in until the surface and the tangent plane become indisting
> 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 = x2 + x
> Find an equation of the tangent plane to the given surface at the specified point. z = ln (x - 2y), (3, 1, 0)
> Determine the signs of the partial derivatives for the function f whose graph is shown. (a). fxy (1, 2) (b). fxy (21, 2)
> Determine the signs of the partial derivatives for the function f whose graph is shown. (a). fxx (21, 2) (b). fyy (21, 2)
> At the beginning of this section we discussed the function I = f (T, H), where I is the heat index, T is the temperature, and H is the relative humidity. Use Table 1 to estimate fT (92, 60) and fH (92, 60). What are the practical interpretations of these
> Cobb and Douglas used the equation P (L, K) = 1.01L0.75K0.25 to model the American economy from 1899 to 1922, where L is the amount of labor and K is the amount of capital. (See Example 14.1.3.) (a). Calculate PL and PK. (b). Find the marginal productivi
> (a). Evaluate t (1, 2, 3). (b). Find and describe the domain of t. Let g(x, y, г) — х'у?г/10 — х — у — г.
> The temperature at a point (x, y) on a flat metal plate is given by T (x, y) = 60/ (1 + x2 + y2), where T is measured in 8C and x, y in meters. Find the rate of change of temperature with respect to distance at the point s2, 1d in (a) the x-direction and
> Show that each of the following functions is a solution of the wave equation utt = a2uxx. (a). u = sin (kx) sin (akt) (b). u = t/ (a2t2 - x2) (c). u = (x – at)6 + (x + at)6 (d). u = sin (x – at) + ln (x + at)
> Level curves are shown for a function f. Determine whether the following partial derivatives are positive or negative at the point P. (a). fx (b). fy (c). fxx (d). fxy (e). fyy 10 8 6 2
> Use the table of values of f (x, y) to estimate the values of fx (3, 2), fx (3, 2.2), and fxy (3, 2). y 1.8 2.0 2.2 2.5 12.5 10.2 9.3 3.0 18.1 17.5 15.9 3.5 20.0 22.4 26.1
> Find the indicated partial derivative(s). f (x, y) = sin (2x + 5y); fyxy
> Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx. u = cos (x2y)
> Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx. u = exy sin y
> (a). Evaluate f (1, 1, 1). (b). Find and describe the domain of f. Let f(x, y, 2) = Vx + vỹ + VE + In(4 – x² – y² – 2ª)
> Find all the second partial derivatives. f (x, y) = x4y - 2x3y2
> Find the first partial derivatives of the function. h (x, y, z, t) = x2y cos (z/t)
> Find fx and fy and graph f, fx, and fy with domains and viewpoints that enable you to see the relationships between them. y f(x, y) = %3D 1 + x?y?
> Find fx and fy and graph f, fx, and fy with domains and viewpoints that enable you to see the relationships between them. f (x, y) = x2y3
> Let (a). Use a computer to graph f. (b). Find fx (x, y) and fy (x, y) when (x, y) ≠(0, 0). (c). Find fx (0, 0) and fy (0, 0) using Equations 2 and 3. (d). Show that fxy (0, 0) = -1 and fyx (0, 0) = 1. (e). Does the result of part (d) c
> If f (x, y) = x (x2 + y2)-3/2 esin (x2 y) find fx (1, 0). [Hint: Instead of finding fx (x, y) first, note that it’s easier to use Equation 1 or Equation 2.]
> (a). How many nth-order partial derivatives does a function of two variables have? (b). If these partial derivatives are all continuous, how many of them can be distinct? (c). Answer the question in part (a) for a function of three variables.
> If f (x, y) = -16 – 4x2 – y2, find fx (1, 2) and fy (1, 2) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.
> a. identify the claim and state H0 and Ha, b. find the critical value(s) and identify the rejection region(s), c. find the standardized test statistic x2, d. decide whether to reject or fail to reject the null hypothesis, and e. interpret the decisio
> Find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D. α = 0.05, d.f.N = 20, d.f.D = 25
> Find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D. α = 0.10, d.f.N = 5, d.f.D = 12
> Find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D. α = 0.01, d.f.N = 12, d.f.D = 10
> a. identify the claim and state H0 and Ha, b. find the critical value and identify the rejection region, c. find the chi-square test statistic, d. decide whether to reject or fail to reject the null hypothesis, and e. interpret the decisi
> Find the equation of the regression line for the data. Then construct a scatter plot of the data and draw the regression line. (Each pair of variables has a significant correlation.) Then use the regression equation to predict the value of y for each of
> Use Table 11 in Appendix B, or perform a hypothesis test using Table 5 in Appendix B to make a conclusion about the correlation coefficient. Refer to the data in Exercise 4. At α = 0.01, is there enough evidence to conclude that there is a
