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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Find an equation of the tangent plane to the graph of ƒ(x, y) = 2x2 − 4xy2 at (−1, 2).
Find the equation of the plane in Figure 10, which is tangent to the graph at (x, y) = (1, 0.8). X 2
Find an equation of the tangent plane at the given point. f(x, y) = xy + xy, (-1,2)
Find an equation of the tangent plane at the given point. f(x, y) = y x (4, -3)
Find an equation of the tangent plane at the given point. f(x, y) = x + y, (4,1)
Find an equation of the tangent plane at the given point.G(u,w) = sin(uw), (π/6, 1)
Find an equation of the tangent plane at the given point. F(r, s) = rs-1/2 + s-, (2,1)
Find an equation of the tangent plane at the given point. g(x,y)= exly, (2, 1)
Find an equation of the tangent plane at the given point. 4 f(x, y) = sech(x - y), (In 4, In 2)
Find an equation of the tangent plane at the given point. (x, y) = ln(4x y), (1,1)
Find the points on the graph of z = 3x2 − 4y2 at which the vector n = (3, 2, 2) is normal to the tangent plane.
Find the points on the graph of z = xy3 + 8y−1 where the tangent plane is parallel to 2x + 7y + 2z = 0.
Find the points on the graph of ƒ(x, y) = 3x2 − xy − y2 at which the tangent plane is horizontal.
Find the points on the graph of ƒ(x, y) = (x + 1)y2 at which the tangent plane is horizontal.
Find the linearization L(x, y) of ƒ(x, y) = x2y3 at (a, b) = (2, 1). Use it to estimate ƒ(2.01, 1.02) and ƒ(1.97, 1.01), and compare with values obtained using a calculator.
Write the Linear Approximation to f (x, y) = x(1 + y)−1 at (a, b) = (8, 1) in the form 7.98 2.02 f(a+h,b+k) f(a,b) + fx(a,b)h + fy(a,b)k Use it to estimate and compare with the value obtained using
Let ƒ(x, y) = x3y−4. Use Eq. (4) to estimate the change Af = f(2.03,0.9) f(2, 1)
Which of the following is a possible value of the gradient ∇ƒ of a function ƒ(x, y) of two variables?(a) 5 (b) (3, 4)(c) (3, 4, 5)
True or false? A differentiable function increases at the rate ΙΙ∇ƒPΙΙ in the direction of ∇ƒP.
Describe the two main geometric properties of the gradient ∇ƒ.
You are standing at a point where the temperature gradient vector is pointing in the northeast (NE) direction. In which direction(s) should you walk to avoid a change in
What is the rate of change of ƒ(x, y) at (0, 0) in the direction making an angle of 45° with the x-axis if ∇ƒ(0, 0) = (2, 4)?
Calculate κ(1) for r(t) = (ln t, t).
Calculate κ(π/4) for r(t) = (tan t, sec t, cos t).
Write the acceleration vector a at the point indicated as a sum of tangential and normal components. r(0) = (cos, sin 20), 0 =
r(t) = (t2, 2t − t2, t), t = 2
At a certain time t0, the path of a moving particle is tangent to the y-axis in the positive direction. The particle’s speed at time t0 is 4 m/s, and its acceleration vector is a = (5, 4, 12).
Give an equation for the osculating circle to y = x2 − x3 at x = 1.
Give an equation for the osculating circle to y = √x at x = 4.
Let r(t) = (cos t, sin t, 2t).(a) Find T,N, and B at the point corresponding to t = π/2. Evaluate T at t = π/2 before finding N and B.(b) Find the equation of the osculating plane at the point
Let r(t) = (ln t, t, t2/2). Find the equation of the osculating plane corresponding to t = 1.
If a planet has zero mass (m = 0), then Newton’s laws of motion reduce to r "(t) = 0 and the orbit is a straight line r(t) = r0 + tv0, where r0 = r(0) and v0 = r'(0) (Figure 1). Show that the area
Suppose the orbit of a planet is an ellipse of eccentricity e = c/a and period T (Figure 2). Use Kepler’s Second Law to show that the time required to travel from A to B is equal to A' e (+) a B 0
The period of Mercury is approximately 88 days, and its orbit has eccentricity 0.205. How much longer does it take Mercury to travel from A to B than from B to A (Figure 2)?
What is the difference between a horizontal trace and a level curve? How are they related?
Describe the trace of ƒ(x, y) = x2 − sin(x3y) in the xz-plane.
Is it possible for two different level curves of a function to intersect? Explain.
Describe the contour map of ƒ(x, y) = x with contour interval 1.
How will the contour maps of with contour interval 1 look different? f(x, y) = x and g(x,y) = 2x
At each point evaluate the function or indicate that the function is undefined there. f(x, y) = x+yx, (1,2), (-1,6), (e, )
At each point evaluate the function or indicate that the function is undefined there. g(x, y) = y x - y (1,3), (3,-3), (2,2)
At each point evaluate the function or indicate that the function is undefined there. h(x, y) = x - y x-y (20, 2), (1, -2), (1, 1)
At each point evaluate the function or indicate that the function is undefined there. k(x,y) = xe, (1,0), (3,3), (0, 12)
At each point evaluate the function or indicate that the function is undefined there. h(x, y, z) = xyz-, (3,7,-2), (3, 2, 1), (4,-4,0)
At each point evaluate the function or indicate that the function is undefined there. w(r, s, t) = r-s sin t (2,2, 3), (n,n,n), (-2,2,)
Sketch the domain of the function. f(x, y) = 12x - 5y
Sketch the domain of the function. f(x, y) = 81-x
Sketch the domain of the function. f(x,y) = In(4x-y)
Sketch the domain of the function. h(x, t) = 1 1+X
Sketch the domain of the function. g(y, z)= 1 z + y
Sketch the domain of the function. f(x, y) = sin X
Sketch the domain of the function.F(I, R) = √IR
Sketch the domain of the function. f(x, y) = cos (x+y)
Describe the domain and range of the function. f(x,y,z) = xz + e
Describe the domain and range of the function. f(x, y, z)=xy + ze/x
Describe the domain and range of the function. P(r, s, t) 16-rs1 =
Describe the domain and range of the function.g(r, s) = cos−1(rs)
What is the difference between D(P, r) and D∗(P, r)?
Suppose that ƒ (x, y) is continuous at (2, 3) and that ƒ(2, y) = y3 for y ≠ 3. What is the value ƒ(2, 3)?
Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). Which of the following statements are true?(a) Q(x, y) is continuous for all (x, y).(b) Q(x, y) is continuous for
Suppose that ƒ(x, 0) = 3 for all x ≠ 0 and ƒ(0, y) = 5 for all y ≠ 0. What can you conclude about lim f(x, y)? (x,y)(0,0)
Evaluate the limit using continuity. lim (x + y) (x,y) (1,2)
Evaluate the limit using continuity. lim (x,y)-(,) y
Evaluate the limit using continuity. lim (xy - 3x^y) (x,y) (-2,1)
Evaluate the limit using continuity. ex lim (x,y)(0,1) x - 4y
Evaluate the limit using continuity. lim tan x cos y (x,y) (7,0)
Evaluate the limit using continuity. lim tan(x - y) (x,y) (2,3)
Evaluate the limit using continuity. et - e-p lim (x,y) (1,1) x+y
Evaluate the limit using continuity. lim In(x - y) (x,y)(1,0)
Assume thatto find the limit. lim f(x, y) = 3, (x,y)-(2,5)* lim g(x, y) = 7 (x,y)-(2,5)
Assume thatto find the limit. lim f(x, y) = 3, (x,y)-(2,5)* lim g(x,y) = 7 (x,y)-(2,5)
Assume thatto find the limit. lim f(x, y) = 3, (x,y)-(2,5)* lim g(x,y) = 7 (x,y)-(2,5)
Assume thatto find the limit. lim f(x, y) = 3, (x,y)-(2,5)* lim g(x,y) = 7 (x,y)-(2,5)
Let ƒ(x, y) = xy/(x2 + y2). Show that ƒ(x, y) approaches zero along the x- and y-axes. Then prove that does not exist by showing that the limit along the line y = x is nonzero. lim f(x,y) (x,y)
Let Set y = mx and show that the resulting limit depends on m, and therefore the does not exist. f(x,y) = 2x + 3y
Prove that does not exist by considering the limit along the x-axis. X lim (x,y)(0,0) x + y
Doesexist? Explain. 12 x + y 2 lim (x,y) (0,0)
Patricia derived the following incorrect formula by misapplying the Product Rule:What was her mistake and what is the correct calculation? ox(+y) = x(2y) + y(2x)
Explain why it is not necessary to use the Quotient Rule to compute Should the Quotient Rule be used to compute a (x + y) axy + 1
Which of the following partial derivatives should be evaluated without using the Quotient Rule? () x y2 + 1 (b) 2 + 1 () 2 2 + 1
What is ƒx, where ƒ(x, y, z) = (sin yz)ez3−z−1 √y?
Assuming the hypotheses of Clairaut’s Theorem are satisfied, which of the following partial derivatives are equal to ƒxxy? (a) fxyx (b) fyyx (c) fxyy (d) fyxx
Use the limit definition of the partial derivative to verify the formulas - = y, dy 2 = 2xy
Use the limit definition of the partial derivative to verify the formulas X 1 2 (1) -= 2 (1) --
Use the Product Rule to compute fx and fy for f(x, y) = (x y)(x y). -
Use the Product Rule to compute fx and fy for f(x, y) = xye* siny.
Use the Quotient Rule to compute ay x + y
Use the Chain Rule to compute - In(u + uv).
Calculate ƒz(2, 3, 1), where ƒ(x, y, z) = xyz.
Explain the relation between the following two formulas (c is a constant): d - sin(cx) = c cos(cx), dx x sin(xy) = y cos(xy)
The plane y = 1 intersects the surface z = x4 + 6xy − y4 in a certain curve. Find the slope of the tangent line to this curve at the point P = (1, 1, 6).
Determine whether the partial derivatives ∂ f /∂x and ∂ f /∂y are positive or negative at the point P on the graph in Figure 6. P Z y
Refer to Figure 7.Estimate ƒx and ƒy at point A. y 2- 0- -2- -4 -20 -4 B -10 -2 70 50 30 / / / 10 0. 10 0 .C -10 2 -30 4 -30 -50 -70 X
Refer to Figure 7.Is ƒx positive or negative at B? y 2- 0- -2- -4 -20 -4 B -10 -2 70 50 30 / / / 10 0. 10 0 .C -10 2 -30 4 -30 -50 -70 X
Refer to Figure 7.Starting at point B, in which compass direction (N, NE, SW, etc.) does ƒ increase most rapidly? y 2- 0- -2- -4 -20 -4 B -10 -2 70 50 30 / / / 10 0. 10 0 .C -10 2 -30 4 -30 -50 -70 X
Refer to Figure 7.At which of A, B, or C is fy the least? y 2- 0- -2- -4 -20 -4 B -10 -2 70 50 30 / / / 10 0. 10 0 .C -10 2 -30 4 -30 -50 -70 X
Compute the first-order partial derivatives. 2+2=2
Compute the first-order partial derivatives. x = 2
Compute the first-order partial derivatives. = 2 ,-2 *y+xy
Compute the first-order partial derivatives.V = πr2h

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