Chapter 17: Functions of Several Variables (Set-2)
When checking lim(x,y)→(a,b)f(x,y)lim(x,y)→(a,b)f(x,y), the most reliable requirement is
A Same value one path
B Exists on x-axis
C Same value every path
D Exists on y-axis
A multivariable limit exists only if the function approaches one unique number for all paths to (a,b)(a,b). If any two paths give different values, the limit fails.
A function is continuous at (a,b)(a,b) only if
A Partial derivatives exist
B Limit equals function value
C Mixed partials are equal
D Gradient is nonzero
Continuity at (a,b)(a,b) needs: f(a,b)f(a,b) defined, lim(x,y)→(a,b)f(x,y)lim(x,y)→(a,b)f(x,y) exists, and the limit equals f(a,b)f(a,b). Partial derivatives are not required.
If f(a,b)f(a,b) is undefined but the limit exists, then
A Not continuous at point
B Continuous by default
C Differentiable at point
D Limit must be zero
Continuity requires the function value to exist and match the limit. If f(a,b)f(a,b) is not defined, continuity fails even if a nice limit exists.
A common quick method to show a limit does not exist is
A Compute second derivatives
B Integrate over region
C Compare two approach paths
D Factor the numerator
If two simple paths to the point (like y=0y=0 and y=xy=x) give different limiting values, the multivariable limit cannot exist.
If substitution gives a finite value for a limit, then
A Limit is guaranteed
B Limit may still fail
C Function is continuous
D Differentiability follows
Direct substitution can mislead in multivariable limits. Even if one path gives a value, other paths might give different results, so existence is not guaranteed.
Directional limits being equal for all directions implies
A Limit always exists
B Continuity always holds
C Differentiability always holds
D Not sufficient always
Equal limits along straight-line directions are necessary, but not always sufficient, because curved paths can still produce different behavior. Full path-independence is required.
A polynomial in x,yx,y is continuous in R2R2 because it is
A Always bounded
B Always differentiable only
C Built from continuous operations
D Always has inverse
Polynomials use only addition and multiplication, which preserve continuity. Therefore, any polynomial function of several variables is continuous everywhere in its domain.
A rational function p/qp/q in x,yx,y is continuous wherever
A Denominator is nonzero
B Numerator is nonzero
C xx is positive
D yy is positive
Rational functions are continuous at all points where they are defined. Discontinuities occur only where the denominator becomes zero, making the expression undefined.
Using polar form near (0,0)(0,0) is most helpful when expression contains
A Only x+yx+y
B x2+y2x2+y2 terms
C Only xyxy
D Only constants
When terms like x2+y2x2+y2 appear, polar substitution x=rcosθ,y=rsinθx=rcosθ,y=rsinθ simplifies them to r2r2, making limit behavior as r→0r→0 clearer.
If after polar conversion the limit depends on θθ, then
A Limit must be zero
B Limit must be one
C Limit does not exist
D Continuity is assured
A true limit at the origin must be the same for all directions. If the expression depends on θθ, different approach angles yield different values, so the limit fails.
In ∂f∂x∂x∂f at (a,b)(a,b), the variable held constant is
A xx fixed
B Both change equally
C Neither is fixed
D yy fixed
Partial derivative with respect to xx treats all other variables as constants. So yy is held fixed while xx changes, measuring the rate of change along the xx-direction.
A first-order partial derivative measures
A Area under surface
B Rate of change direction
C Curve length on surface
D Volume under surface
First partial derivatives describe how the function changes when one variable changes slightly, keeping others constant. It is the multivariable version of slope in a chosen direction.
The mixed partial fxyfxy means
A Differentiate y then y
B Differentiate x then y
C Differentiate x then x
D Integrate then differentiate
fxyfxy means take ∂f/∂x∂f/∂x first, then differentiate that result with respect to yy. Under smoothness conditions, it equals fyxfyx.
Clairaut’s theorem is used mainly to
A Prove continuity always
B Swap mixed partial order
C Compute Jacobian only
D Find limits quickly
If mixed partial derivatives are continuous near a point, then fxy=fyxfxy=fyx. This allows switching differentiation order to simplify computation.
The gradient of f(x,y)f(x,y) is
A ⟨fx,fy⟩⟨fx,fy⟩
B ⟨f,x⟩⟨f,x⟩
C ⟨x,y⟩⟨x,y⟩
D ⟨fxx,fyy⟩⟨fxx,fyy⟩
The gradient is the vector of first partial derivatives. It points in the direction of steepest increase of the function, and its magnitude gives the maximum rate of increase.
Directional derivative DufDuf requires uu to be
A Any nonzero vector
B A zero vector
C A unit vector
D A tangent vector
The directional derivative uses a unit vector so the value reflects direction only, not vector length. Otherwise, scaling the direction vector would incorrectly scale the derivative.
Formula for directional derivative in direction uu is
A ∇f×u∇f×u
B ∇f⋅u∇f⋅u
C f⋅uf⋅u
D f/uf/u
Directional derivative is the dot product of the gradient with the unit direction vector. This projects the gradient onto the chosen direction to get the rate of change.
For z=f(x,y)z=f(x,y), the tangent plane at (a,b)(a,b) uses
A fxx,fyyfxx,fyy values
B Only Jacobian value
C Only Euler theorem
D fx,fyfx,fy values
The tangent plane comes from linear approximation: z≈f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)z≈f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b). Only first partial derivatives are needed.
Total differential dfdf for f(x,y)f(x,y) is
A f dx+f dyfdx+fdy
B fxdx+fydyfxdx+fydy
C dx+dydx+dy
D fxdy+fydxfxdy+fydx
The total differential gives the best linear estimate of change in ff for small changes in xx and yy. It is widely used in approximation and error estimation.
If x=x(t),y=y(t)x=x(t),y=y(t), then dzdtdtdz for z=f(x,y)z=f(x,y) equals
A fxx′+fyy′fxx′+fyy′
B fx+fyfx+fy
C x′+y′x′+y′
D fxytfxyt
Chain rule adds contributions through both variables: dz/dt=fx dx/dt+fy dy/dtdz/dt=fxdx/dt+fydy/dt. It explains how zz changes as tt changes indirectly via x(t),y(t)x(t),y(t).
A function f(x,y)f(x,y) is homogeneous of degree nn if
A f(tx,ty)=ff(tx,ty)=f always
B f(tx,ty)=tnff(tx,ty)=tnf
C f(tx,ty)=t+ff(tx,ty)=t+f
D f(tx,ty)=0f(tx,ty)=0
Homogeneity means scaling inputs by tt scales output by tntn. The exponent nn is the degree. This helps identify scaling properties and simplify derivatives.
Euler’s theorem for degree nn in two variables states
A fx+fy=nfx+fy=n
B fxx+fyy=0fxx+fyy=0
C xfx+yfy=nfxfx+yfy=nf
D x+y=nx+y=n
For a homogeneous function of degree nn, a weighted sum of first partial derivatives equals nn times the function. It is useful for quick checks and simplifications.
If a function is homogeneous of degree 1, then Euler gives
A xfx+yfy=0xfx+yfy=0
B xfx+yfy=fxfx+yfy=f
C xfx+yfy=2fxfx+yfy=2f
D xfx+yfy=1xfx+yfy=1
Degree 11 means output scales linearly with input scaling. Substituting n=1n=1 into Euler’s theorem yields xfx+yfy=fxfx+yfy=f, a common identity in applications.
To verify homogeneity quickly, the best step is to
A Replace x,yx,y by tx,tytx,ty
B Compute Hessian first
C Find Jacobian first
D Integrate over region
Substitute scaled variables and factor out tntn. If a single power of tt factors from the entire expression, the function is homogeneous and the exponent is its degree.
For f(x,y,z)f(x,y,z) homogeneous of degree nn, Euler form is
A fx+fy+fz=nfx+fy+fz=n
B fxx+fyy+fzz=0fxx+fyy+fzz=0
C xfx+yfy+zfz=nfxfx+yfy+zfz=nf
D x+y+z=nx+y+z=n
The theorem extends naturally to three variables: multiply each partial derivative by its variable and add. If the function is homogeneous of degree nn, the sum equals nfnf.
The Jacobian ∂(u,v)∂(x,y)∂(x,y)∂(u,v) is
A Sum of partials
B Product of variables
C Laplacian value
D Determinant of partials
It is the determinant of the matrix formed by ∂u/∂x,∂u/∂y,∂v/∂x,∂v/∂y∂u/∂x,∂u/∂y,∂v/∂x,∂v/∂y. It measures local area scaling under the mapping.
If u=u(x,y)u=u(x,y) and v=v(x,y)v=v(x,y), then the Jacobian matrix is
A (uvxy)(uxvy)
B (uxuyvxvy)(uxvxuyvy)
C (xuyv)(xyuv)
D (uxvxuyvy)(uxuyvxvy)
Rows correspond to functions u,vu,v and columns correspond to variables x,yx,y. The Jacobian determinant is computed from this matrix and describes local transformation behavior.
If ∂(u,v)∂(x,y)≠0∂(x,y)∂(u,v)=0 at a point, then mapping is
A Globally periodic
B Always linear
C Locally invertible
D Always continuous only
A nonzero Jacobian at a point indicates the transformation has a local inverse nearby (basic invertibility condition). It means the mapping does not collapse area to zero at that point.
For inverse transformations, a basic relation is
A Juv/xy+Jxy/uv=1Juv/xy+Jxy/uv=1
B Juv/xy−Jxy/uv=0Juv/xy−Jxy/uv=0
C Juv/xy Jxy/uv=0Juv/xyJxy/uv=0
D Juv/xy Jxy/uv=1Juv/xyJxy/uv=1
When an inverse exists and derivatives behave nicely, the Jacobians are reciprocals. Their product equals 1, expressing how scaling in one transformation reverses under the inverse map.
For polar coordinates x=rcosθ,y=rsinθx=rcosθ,y=rsinθ, ∣∂(x,y)∂(r,θ)∣∂(r,θ)∂(x,y) equals
A 1/r1/r
B rr
C r2r2
D sinθsinθ
The Jacobian determinant for polar conversion is rr, so area element becomes dA=r dr dθdA=rdrdθ. This factor accounts for stretching of grid spacing away from the origin.
A “singular Jacobian” at a point typically means
A Determinant becomes one
B Matrix is symmetric
C Mixed partials unequal
D Determinant becomes zero
Singular Jacobian means the determinant vanishes. The mapping locally collapses area/volume, and invertibility can fail. It often signals problematic points in variable transformation.
For u=x+yu=x+y, v=x−yv=x−y, ∂(u,v)∂(x,y)∂(x,y)∂(u,v) equals
A 00
B −2−2
C 11
D 22
Compute ux=1,uy=1,vx=1,vy=−1ux=1,uy=1,vx=1,vy=−1. Jacobian =(1)(−1)−(1)(1)=−2=(1)(−1)−(1)(1)=−2. The negative sign indicates orientation reversal; magnitude gives area scaling.
A Jacobian is most directly used in
A Long division
B Change of variables
C Partial fractions
D Binomial expansion
In double and triple integrals, Jacobians adjust for how area/volume elements transform. They are essential when switching coordinates like Cartesian to polar, cylindrical, or spherical.
In three variables, ∂(u,v,w)∂(x,y,z)∂(x,y,z)∂(u,v,w) is a
A 2×2 determinant
B 3×3 determinant
C 1×1 determinant
D 4×4 determinant
The Jacobian for three variables is the determinant of the 3×3 matrix of first partial derivatives. It measures local volume scaling under transformation between (x,y,z)(x,y,z) and (u,v,w)(u,v,w).
A functional dependence test often checks whether a certain Jacobian is
A Identically one
B Always negative
C Identically zero
D Always increasing
If functions are dependent, the transformation loses rank and the Jacobian determinant can vanish. A zero Jacobian (in the right setup) indicates lack of independence among variables.
Differentiability of f(x,y)f(x,y) at a point implies
A Discontinuity at point
B Jacobian equals one
C Euler theorem holds
D Continuity at point
Differentiability gives a linear approximation near the point, which forces the function to approach its value smoothly. Thus differentiability implies continuity, but continuity alone may not imply differentiability.
A critical point of f(x,y)f(x,y) usually satisfies
A f=0f=0 only
B fx=0,fy=0fx=0,fy=0
C fxy=0fxy=0 only
D Jacobian equals zero
Interior candidate points for maxima, minima, or saddles occur where both first partial derivatives vanish (or do not exist). These points must then be classified using further tests.
For the second derivative test, the discriminant is
A D=fxfyD=fxfy
B D=fxxfyy−(fxy)2D=fxxfyy−(fxy)2
C D=fxx+fyyD=fxx+fyy
D D=fxy+fyxD=fxy+fyx
At a critical point, DD summarizes curvature. Its sign helps classify behavior: D>0D>0 suggests min/max depending on fxxfxx, while D<0D<0 indicates a saddle.
If D>0D>0 and fxx>0fxx>0, then point is
A Local maximum
B Local minimum
C Saddle point
D No conclusion
D>0D>0 means curvature is consistent in nearby directions. If fxx>0fxx>0, the surface bends upward like a bowl, giving a local minimum at the critical point.
If D>0D>0 and fxx<0fxx<0, then point is
A Local maximum
B Local minimum
C Saddle point
D No conclusion
With D>0D>0, the surface has same-type curvature in directions. If fxx<0fxx<0, the surface bends downward near the point, producing a local maximum.
If D<0D<0 at a critical point, then point is
A Local minimum
B Local maximum
C Saddle point
D Global minimum
D<0D<0 indicates opposite curvature in different directions, like up in one direction and down in another. This is the hallmark of a saddle point in two-variable functions.
If D=0D=0 in second derivative test, then
A Must be minimum
B Must be maximum
C Must be saddle
D Test is inconclusive
When D=0D=0, quadratic terms do not fully determine local shape. You must inspect higher-order terms or use another method to classify the critical point.
Lagrange multipliers are mainly used for
A Finding simple limits
B Constrained optimization
C Solving linear systems
D Expanding polynomials
This method finds maxima/minima of f(x,y)f(x,y) subject to constraints like g(x,y)=cg(x,y)=c. It uses the condition ∇f=λ∇g∇f=λ∇g at optimal points.
On the level curve f(x,y)=cf(x,y)=c, the gradient ∇f∇f is
A Tangent to curve
B Perpendicular to curve
C Parallel to x-axis
D Always zero
Along a level curve, ff stays constant, so movement tangent to the curve causes zero change. The gradient points toward greatest increase, so it must be normal to the level curve.
For surface F(x,y,z)=0F(x,y,z)=0, a normal direction at a point is given by
A ∇F∇F vector
B ∇f∇f vector
C Any tangent vector
D Position vector
The gradient of FF is perpendicular to level surfaces F=constantF=constant. Therefore, on F(x,y,z)=0F(x,y,z)=0, ∇F∇F provides the normal direction (when ∇F≠0∇F=0).
The equation of normal line to z=f(x,y)z=f(x,y) at (a,b)(a,b) uses
A Only f(a,b)f(a,b) value
B Direction from surface normal
C Only fxxfxx value
D Only Jacobian value
A normal line is a line through the point on the surface with direction equal to a normal vector. For graphs z=f(x,y)z=f(x,y), this relates to the normal of the tangent plane.
Total differential is most useful for
A Exact global maximum
B Solving quadratic roots
C Small change approximation
D Computing area exactly
Total differential approximates change in ff from small changes in inputs: Δf≈fxΔx+fyΔyΔf≈fxΔx+fyΔy. It is a practical tool for quick estimates and error bounds.
A basic condition often used in implicit differentiation F(x,y)=0F(x,y)=0 is
A F=0F=0 never
B Fx=0Fx=0 always
C y=0y=0 always
D Fy≠0Fy=0
If Fy≠0Fy=0 near a point, yy can often be treated as a differentiable function of xx locally. Then dy/dx=−Fx/Fydy/dx=−Fx/Fy follows by implicit differentiation.
Continuity of a composite function f(g(x,y),h(x,y))f(g(x,y),h(x,y)) is ensured if
A Only outer is continuous
B All pieces are continuous
C Only inner is continuous
D Only partials exist
If gg and hh are continuous and f(u,v)f(u,v) is continuous at the corresponding point, then the composite f(g(x,y),h(x,y))f(g(x,y),h(x,y)) is continuous by continuity-preserving composition rules.
A standard linear approximation near (a,b)(a,b) for f(x,y)f(x,y) is
A f(a,b)+fxΔx+fyΔyf(a,b)+fxΔx+fyΔy
B f(a,b)+fxxΔx2f(a,b)+fxxΔx2
C f(a,b)+fyyΔy2f(a,b)+fyyΔy2
D f(a,b)+ΔxΔyf(a,b)+ΔxΔy
The best first-order approximation uses the tangent plane: f(a+Δx,b+Δy)≈f(a,b)+fx(a,b)Δx+fy(a,b)Δyf(a+Δx,b+Δy)≈f(a,b)+fx(a,b)Δx+fy(a,b)Δy. It is widely used in multivariable estimation