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Find (x,y)→(−1,2)lim​x2+y2−22x2+y2−3​ (x,y)→(−1,2)lim​x2+y2−22x2+y2−3​= (Type an integer or a simplified fraction.)
Find (x,y)→(0,0)lim​x−2e3ysin(−5x)​ (x,y)→(0,0)lim​x−2e3ysin(−5x)​= (Type an integer or a simplified fraction.)
Use the two-path test to prove that the following limit does not exist. (x,y)→(0,0)lim​y4+x2y4−2x2​ What value does f(x,y)=y4+x2y4−2x2​ approach as (x,y) approaches (0,0) along the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. f(x,y) approaches (Simplify your answer.) B. f(x,y) approaches ∞. C. f(x,y) approaches −∞. D. f(x,y) has no limit as (x,y) approaches (0,0) along the x-axis.
Find the limit of f as (x,y)→(0,0) or show that the limit does not exist. Consider converting the functio f(x,y)=x2+y2x3−xy2​ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (x,y)→(0,0)lim​f(x,y)= (Simplify your answer.) B. The limit does not exist.
For the function f(x,y)=6x4−4y−5, find ∂x∂f​ and ∂y∂f​. ∂x∂f​=
Find ∂x∂f​ and ∂y∂f​ f(x,y)=4x2+y2​ ∂x∂f​= (Type an exact answer, using radicals as needed.)
For the function f(x,y)=sin8(5x−3y), find ∂x∂f​ and ∂y∂f​. ∂x∂f​=
Find ∂f/∂x and ∂f/∂y. f(x,y)=3x3y ∂x∂f​=
Find all the second-order partial derivatives of the following function. w=5x2tan(4xy) ∂x2∂2w​=
f(x,y)=e−9ysin9x Find the second-order partial derivatives of f(x,y) with respect to x and y, respectively. ∂x2∂2f​=∂y2∂2f​=□
z=3tan−1(yx​),x=ucosv,y=usinv,(u,v)=(5.4,3π​) Express ∂x∂z​ and ∂y∂z​ as functions of x and y. ∂x∂z​= (Type an exact answer.) 
Draw a dependency diagram, and write a chain rule formula for dp∂w​ and dqdw​, where w=f(r,s,t),r=h(p,q), s=k(p,q), and t=g(p,q). Choose the correct dependency diagram for ∂p∂w​.
Assuming −5x3−y2+4xy=0 defines y as a differentiable function of x, use the theorem dxdy​=−Fy​Fx​​ to find dxdy​ at the point (−1,1). dxdy​∣∣​(−1,1)​= (Type an integer or a simplified fraction.)
If the equation F(x,y,z)=0 determines z as a differentiable ft ∂x∂z​=−Fz​Fx​​ and ∂y∂z​=−Fz​Fy​​ Use these equations to find the values of ∂z/∂x and ∂z/∂y z3−2xy+4yz+y3−131=0,(1,4,3) dxdz​∣∣​(1,4,3)​= (Type an integer or a simplified fraction.)
Find ∂u∂z​ and ∂v∂z​ when u=ln3,v=1, if z=10cot−1x, and x=eu+lnv ∂u∂z​∣∣​u=ln3,v=1​= (Simplify your answer.)
Assume that z=f(w),w=g(x,y),x=4r3−s2, and y=res. If gx​(4,1)=−3,gy​(4,1)=4,f′(8)=−3, and g(4,1)=8, find the following. ∂r∂z​∣∣​r=1,s=0​ and ∂s∂z​∣∣​r=1,s=0​ Calculate ∂r∂z​∣∣​r=1,s=0​ and ∂s∂z​∣∣​r=1,s=0​ ∂r∂z​∣∣​r=1,s=0​=∂s∂z​∣∣​r=1,s=0​=​
Find ∂u∂z​ when u=−1,v=0, if z=sin(xy)+xsin(y),x=u2+3v2, and y=3uv. ∂u∂z​∣∣​u=−1,v=0​= (Simplify your answer.)
Find the gradient of the function g(x,y)=xy2 at the point (3,−1). Then sketch the gradient together with the level curve that passes through the point. First find the gradient vector at (3,−1). ∇g(3,−1)=i+(∣j (Simplify your answers.)
Find the gradient of f(x,y,z)=(x2+y2+z2)−1/2+ln(xyz) at the point (−1,2,−2) ∇f∣(−1,2,−2)​=∣∣i+1∣j+∣∣∣k
Find the derivative of the function at P0​ in the direction of A. f(x,y)=4xy−5y2,P0​(7,3),A=−4i+8j (DA​f)(7,3)= (Type an exact answer, using radicals as needed.)
Find the derivative of the function at P0​ in the direction of A. f(x,y,z)=xy+yz+zx,P0​(3,−3,2),A=6i+2j−3k (DA​)(3,−3,2)​= (Simplify your answer.)
f(x,y)=x2+xy+y2,P0​(1,4) The direction in which the given function f(x,y)=x2+xy+y2 increases most rapidly at P0​(1,4) is u=i+(j. (Type exact answers, using radicals as needed.)
f(x,y,z)=4ln(xy)+ln(yz)+ln(xz),P0​(1,1,1) In which direction does the function increase most rapidly? u=(i+1∣j+1∣k (Type exact answers, using radicals as needed.)
Find the derivative of the function at P0​ in the direction of A. f(x,y,z)=−9excos(yz),P0​(0,0,0),A=i+3j+3k (DA​)(0,0,0)​= (Type an exact answer, using radicals as needed.)
Let f(x,y)=x2−xy+y2−y. Find the directions u and the values of Du​f(1,−1) for which the following is true. a. Du​f(1,−1) is largest b. Du​f(1,−1) is smallest c. Du​f(1,−1)=0 d. Du​f(1,−1)=4 e. Du​f(1,−1)=−3 a. Find the direction u and the value of Du​f(1,−1) for which Du​f(1,−1) is largest. u=i+jj
In what directions is the derivative of f(x,y)=xy+y2 at P(1,6) equal to zero? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. u= (Simplify your answer. Use a comma to separate answers as needed. Type your answer in terms of i and j.) B. There is no solution.
Is there a direction u in which the rate of change of f(x,y)=x2−3xy+4y2 at P(1,2) equals 14 ? Give reasons for your answer. Choose the correct answer below. A. Yes. The given rate of change is larger than the minimum rate of change and smaller than the maximum rate of change. B. No. The given rate of change is larger than the maximum rate of change. C. No. The given rate of change is smaller than the minimum rate of change.
Find a parametric equation for the line that is perpendicular to the graph of the equation x2+4y2+3z2=17 at the point (1,1,2). r(t)=1)i+()j+()k (Type expressions using t as the variable.)