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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Describe the relation between the vector J = r × r and the rate at which the radial vector sweeps out area.
Find r(t) and v(t) given a(t) and the initial velocity and position. a(t) = (e, 2t, t + 1), v(0) = (1,0,1), r(0) = (2, 1,1)
Find r(t) and v(t) given a(t) and the initial velocity and position. a(t) = (t, 4), v(0) = (3,-2), r(0) = (0,0)
Find v(t) given a(t) and the initial velocity.a(t) = t2k, v(0) = i − j
Find v(t) given a(t) and the initial velocity.a(t) = k, v(0) = i
Find v(t) given a(t) and the initial velocity.a(t) = (et, 0, t + 1), v(0) = (1, −3,√2)
Find v(t) given a(t) and the initial velocity. a(t) = (t, 4), v(0) = (-2)
The paths r(t) =(t2, t3) and r1(t) = (t4, t6) trace the same curve, and r1(1) = r(1). Do you expect either the velocity vectors or the acceleration vectors of these paths at t = 1 to point in the
Sketch the path r(t) = (t2, t3) together with the velocity and acceleration vectors at t = 1.
Sketch the path r(t) = 1 − t2, 1 − t for −2 ≤ t ≤ 2, indicating the direction of motion. Draw the velocity and acceleration vectors at t = 0 and t = 1.
Find a(t) for a particle moving around a circle of radius 8 cm at a constant speed of v = 4 cm/s (see Example 4). Draw the path, and on it, draw the acceleration vector at t = π/4. EXAMPLE 4 Uniform
Calculate the velocity and acceleration vectors and the speed at the time indicated. S 7(0) = (1 +51 +38 ) = r(s) S S = 2
Calculate the velocity and acceleration vectors and the speed at the time indicated. r(0) = (sin 0, cos 0, cos 30), 0 =
Calculate the velocity and acceleration vectors and the speed at the time indicated.r(t) = etj − cos(2t)k, t = 0
Calculate the velocity and acceleration vectors and the speed at the time indicated. r(t) = (t, 1-t, 4t), t = 1
Draw the vectors r(2 + h) − r(2) and for h = 0.5 for the path in Figure 10. Draw v(2) (using a rough estimate for its length). r(2+h)-r(2) h
Use the table to calculate the difference quotients for h = −0.2, −0.1, 0.1, 0.2. Then estimate the velocity and speed at t = 1. r(1+h)-r(1) h
Two cars are racing around a circular track. If, at a certain moment, both of their speedometers read 110 mph. then the two cars have the same (choose one):(a) aT (b) aN
What is the length of the acceleration vector of a particle traveling around a circle of radius 2 cm with constant velocity 4 cm/s?
If a particle travels along a straight line, then the acceleration and velocity vectors are (choose the correct description):(a) Orthogonal (b) Parallel
Use the decomposition of acceleration into tangential and normal components to explain the following statement: If the speed is constant, then the acceleration and velocity vectors are orthogonal.
Two objects travel to the right along the parabola y = x2 with nonzero speed. Which of the following statements must be true?(a) Their velocity vectors point in the same direction.(b) Their velocity
For a particle in uniform circular motion around a circle, which of the vectors v(t) or a(t) always points toward the center of the circle?
If a particle travels with constant speed, must its acceleration vector be zero? Explain.
Find the curvature of the plane curve at the point indicated.y = t4, t = 2
Find the curvature of the plane curve at the point indicated.y = cos x, x = 0
find the curvature of the plane curve at the point indicated.y = et, t = 3
Use Eq. (3) to evaluate the curvature at the given point.r(t) = (cosh t, sinh t, t), t = 0 k(t) = |r'(z) x r"(t)|| ||r'()||
Use Eq. (3) to evaluate the curvature at the given point. k(t) = |r' (t) x r' (t)|| ||r'()||
Use Eq. (3) to evaluate the curvature at the given point. k(t) = |r' (t) x r' (t)|| ||r'()||
Use Eq. (3) to evaluate the curvature at the given point. k(t) = |r' (t) x r' (t)|| ||r'()||
Use Eq. (3) to calculate the curvature function κ(t).r(t) = (t−1, 1, t) K(t) = |r'(t) x r"(t)|| ||r'()||
Use Eq. (3) to calculate the curvature function κ(t).r(t) = (4t + 1, 4t − 3, 2t) K(t) = |r' (t) x r"(t)|| ||r'()||
Use Eq. (3) to calculate the curvature function κ(t).r(t) = (4 cos t, t, 4 sin t) k(t) = |r'(t) x r"(t)|| ||r'()||
Use Eq. (3) to calculate the curvature function κ(t).r(t) = (1, et, t) k(t) = |r' (t) x r"(t)|| ||r'()||
Calculate r '(t) and T(t), and evaluate T(1).r(t) = (et, e−t, t2)
Calculate r '(t) and T(t), and evaluate T(1).r(t) = (cos πt, sin πt, t)
Calculate r '(t) and T(t), and evaluate T(1).r(t) = (1 + 2t, t2, 3 − t2)
Calculate r '(t) and T(t), and evaluate T(1).r(t) = (3 + 4t, 3 − 5t, 9t)
Calculate r '(t) and T(t), and evaluate T(1). r(t) = (e, 1)
Calculate r '(t) and T(t), and evaluate T(1). r(t) = (41,9t)
Compute the angle between the two planes, defined as the angle θ (between 0 and π) between their normal vectors (Figure 10).Planes with normals n1 = (1, 0, 1), n2 = (−1, 1, 1) n 112 n2 0 n PL P
Suppose that a plane P with normal vector n and a line L with direction vector v both pass through the origin and that n · v = 0. Which of the following statements is correct?(a) L is contained in
Which of the following planes contains the z-axis?(a) z = 1(b) x + y = 1 4(c) x + y = 0
To which coordinate plane is the plane y = 1 parallel?
The vector k is normal to which of the following planes?(a) x = 1(b) y = 1(c) z = 1
What is the equation of the plane parallel to 3x + 4y − z = 5 passing through the origin?
This exercise gives another proof of the relation between the dot product and the angle θ between two vectors v = (a1, b1) and w = (a2, b2) in the plane. Observe that v = ΙΙvΙΙ (cos θ1, sin
(a) Divide the power series in Exercise 41 by 3x2 to obtain a power series for h(x) = 1/(1 − x3)2 and use the Ratio Test to show that the radius of convergence is 1.Data From Exercise
Show that S2 also gives the exact value for ∫ba x3 dx and conclude, as in Exercise 63, that SN is exact for all cubic polynomials. Show by counterexample that S2 is not exact for integrals of
According to Planck’s Radiation Law, the amount of electromagnetic energy with frequency between ν and ν + Δν that is radiated by a so-called black body at temperature T is proportional to
Let b > 0 and let ƒ(x) = F(xG(b)) with F as in Exercise 117. Use Exercise 116 (f) to prove that ƒ(r) = br for every rational number r. This gives us a way of defining bx for irrational x, namely
A thin silver wire has length L = 18 cm when the temperature is T = 30°C. Estimate ΔL when T decreases to 25°C if the coefficient of thermal expansion is k = 1.9 × 10−5°C−1 (see Example 3).
What is the radius of curvature at P if κP = 9?
What is the radius of curvature of a circle of radius 4?
Refer to the function ƒ whose graph is shown in Figure I.For which value of h is ƒ(0.7 + h) − ƒ(0.7)/h equal to the slope of the secant line between the points where x = 0.7 and x = 1.1? 6- 5432
What is the curvature at a point where T(s) = (1, 2, 3) in an arc length parametrization r(s)?
What is the curvature of r(t) = (2 + 3t, 7t, 5 − t)?
Which has larger curvature, a circle of radius 2 or a circle of radius 4?
What is the curvature of a circle of radius 4?
What is the unit tangent vector of r(t) if the underlying curve is a line with direction vector w = (2, 1, −2) and x is decreasing along r(t)?
Find the speed at the given value of t. r(t) = (sin 3t, cos 4t, cos 5t), t =
Find the speed at the given value of t. r(t) (e-3, 12, 3t-), t = 3 =
Find the speed at the given value of t. ) = (t, Int, (Int)), t = 1
Find the speed at the given value of t. r(t) = (-1,21, -3t), t = 2
Find the speed at the given value of t. r(t) = (2t + 3,4t - 3,5-t), t = 4
Compute the arc length function s(t) = ||r'(u)|| du for the given value of a.
Compute the arc length function s(t) = ||r' (u)|| du for the given value of a.
The curve shown in Figure 5(B) is parametrized by r(t) = (2 cos t, 2 sin t, cos(19t)) for 0 ≤ t ≤ 2π. Approximate its length. X' + 2 2 Z (B)
The curve shown in Figure 5(A) is parametrized by r(t) = (cos(7t), sin(7t), 2 cos t) for 0 ≤ t ≤ 2π. Approximate its length. -2 24 -24 (A) 2 NJ 24 ~ (B)
Compute the length of the curve traced by r(t) over the given interval. r(t) = ti + 2tj + (t-3)k, 0t2
Compute the length of the curve traced by r(t) over the given interval. r(t)= (t cos t, t sint, 3t), 0t 2
Compute the length of the curve traced by r(t) over the given interval. r(t) = (2t + 1,21 - 1,), 0t2
Compute the length of the curve traced by r(t) over the given interval. r(t) = (1,413/2, 213/2), 0t3
Compute the length of the curve traced by r(t) over the given interval. r(t) = (cost, sint, 13/2), 0t 2
Compute the length of the curve traced by r(t) over the given interval. r(t) = (2t, ln t,t), 1t4
Compute the length of the curve traced by r(t) over the given interval.r(t) = (2t, 2 − 4t, 5), 5≤ t ≤ 10
Compute the length of the curve traced by r(t) over the given interval.r(t) =(3t, 4t − 3, 6t + 1), 0≤ t ≤ 3
What is the length of the path traced by r(t) for 4 ≤ t ≤ 10 if r(t) is an arc length parametrization?
Starting at the origin, a mosquito flies along a parabola with speed v(t) = t2. Let L(t) be the total distance traveled at time t.(a) How fast is L(t) changing at t = 2?(b) Is L(t) equal to the
Two cars travel in the same direction along the same roller coaster (at different times). Which of the following statements about their velocity vectors at a given point P on the roller coaster are
At a given instant, a car on a roller coaster has velocity vector v = (25, −35, 10) (in miles per hour). What would the velocity vector be if the speed were doubled? What would it be if the car’s
State whether the following derivatives of vector-valued functions r1(t) and r2(t) are scalars or vectors: d (a) -r(t) dt (b) dt (r(t) r(t)) - (c) (r(t) x r(t)) dt
Indicate whether the statement is true or false, and if it is false, provide a correct statement.The derivative of the cross product is the cross product of the derivatives.
Indicate whether the statement is true or false, and if it is false, provide a correct statement.The derivative of a vector-valued function is the slope of the tangent line, just as in the scalar
Indicate whether the statement is true or false, and if it is false, provide a correct statement.The terms “velocity vector” and “tangent vector” for a path r(t) mean the same thing.
Indicate whether the statement is true or false, and if it is false, provide a correct statement.The integral of a vector-valued function is obtained by integrating each component.
Indicate whether the statement is true or false, and if it is false, provide a correct statement.The derivative of a vector-valued function is defined as the limit of a difference quotient, just as
State the three forms of the Product Rule for vector-valued functions.
Which three of the following vector-valued functions parametrize the same space curve? (a) (-2+ cost)i + 9j + (3- sint)k (c) (-2+cos3t)i + 9j + (3 sin 3t)k (e) (2 + cost)i + 9j + (3 + sin t)k (b) (2
How do the paths r1(t) = (cos t, sin t) and r2(t) = (sin t, cos t) around the unit circle differ?
What is the center of the circle with the following parametrization?r(t) = (−2 + cos t)i + 2j + (3 − sin t)k
What is the projection of r(t) = ti + t4j + etk onto the xz-plane?
Which one of the following does not parametrize a line? (a) r(t) (8 t, 2t, 3t) = (b) r(t) = ti - 7tj + tk (c) r3(t) = (8-413,2+51,91)
Let c be a scalar, a and b be vectors, and X = (x, y, z). Show that the equation (X − a) · (X − b) = c2 defines a sphere with center m = 1/2 (a + b) and radius R, where R2 = c2 + ΙΙ1/2 (a −
Show that the spherical equation cot ϕ = 2 cos θ + sin θ defines a plane through the origin (with the origin excluded). Find a normal vector to this plane.
Describe how the surface with spherical equationdepends on the constant A. p(1 + A cos ) = 1
Sketch the graph of the spherical equation ρ = 2 cos θ sin ϕand write the equation in rectangular coordinates.
Show that the cylindrical equationis a hyperboloid of one sheet. r(1-2 sin0) + z = 1
Write the surface x2 + y2 − z2 = 2 (x + y) as an equation r = ƒ(θ, z) in cylindrical coordinates.

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