Chapter 17: Functions of Several Variables (Set-4)
For f(x,y)=x2−y2x2+y2f(x,y)=x2+y2x2−y2, as (x,y)→(0,0)(x,y)→(0,0) the limit
A Equals 0
B Does not exist
C Equals 1
D Equals −1
Along y=0y=0, f=1f=1. Along x=0x=0, f=−1f=−1. Since two paths give different values, the limit at (0,0)(0,0) cannot be unique, so it does not exist.
For f(x,y)=x2yx2+y2f(x,y)=x2+y2x2y, the limit at (0,0)(0,0) is
A Infinity
B 1
C Does not exist
D 0
Use polar: x=rcosθ,y=rsinθx=rcosθ,y=rsinθ. Then f=r2cos2θ⋅rsinθr2=rcos2θsinθ→0f=r2r2cos2θ⋅rsinθ=rcos2θsinθ→0 as r→0r→0, independent of θθ.
If ∣f(x,y)∣≤x2+y2∣f(x,y)∣≤x2+y2 near (0,0)(0,0), then the limit of ff at origin is
A 1
B Does not exist
C 0
D Depends on path
Since 0≤∣f(x,y)∣≤x2+y20≤∣f(x,y)∣≤x2+y2 and x2+y2→0x2+y2→0 as (x,y)→(0,0)(x,y)→(0,0), the squeeze principle forces ∣f∣→0∣f∣→0, hence f→0f→0.
A function can be continuous at a point even if
A Limit does not exist
B Partial derivatives fail
C Value is undefined
D Two paths differ
Continuity needs only that the limit exists and equals the function value. Partial derivatives may fail at that point, yet the function can still be perfectly continuous.
If ff is differentiable at (a,b)(a,b), then ff is
A Continuous at (a,b)(a,b)
B Discontinuous at (a,b)(a,b)
C Unbounded at (a,b)(a,b)
D Nonexistent at (a,b)(a,b)
Differentiability gives a linear approximation with small error, which forces function values to approach f(a,b)f(a,b) as (x,y)→(a,b)(x,y)→(a,b). Hence differentiability always implies continuity.
Equal limits along all straight lines through origin are
A Always sufficient
B Same as continuity
C Same as differentiability
D Not sufficient always
Straight lines test only a subset of paths. Curved paths can still give different limiting values. Therefore equal line-limits are necessary but may not be sufficient for full limit existence.
In R3R3, continuity at (a,b,c)(a,b,c) uses distance
A ∣x−a∣+∣y−b∣∣x−a∣+∣y−b∣
B (x−a)2+(y−b)2+(z−c)2(x−a)2+(y−b)2+(z−c)2
C ∣x−a∣∣x−a∣ only
D ∣z−c∣∣z−c∣ only
The natural notion of closeness in R3R3 is Euclidean distance. Epsilon–delta continuity uses this distance to ensure function values stay within a desired tolerance.
For f(x,y)=x2+y2f(x,y)=x2+y2, the gradient at (1,2)(1,2) is
A ⟨1,2⟩⟨1,2⟩
B ⟨4,2⟩⟨4,2⟩
C ⟨2,4⟩⟨2,4⟩
D ⟨0,0⟩⟨0,0⟩
∇f=⟨fx,fy⟩=⟨2x,2y⟩∇f=⟨fx,fy⟩=⟨2x,2y⟩. At (1,2)(1,2), it becomes ⟨2,4⟩⟨2,4⟩. This points toward steepest increase of ff there.
Directional derivative is zero in a direction perpendicular to
A Position vector
B Tangent plane
C Normal line
D Gradient vector
Duf=∇f⋅uDuf=∇f⋅u. If uu is perpendicular to ∇f∇f, the dot product is zero, meaning no first-order change in that direction.
For z=f(x,y)z=f(x,y), tangent plane at (a,b)(a,b) has equation
A z=f+fx(x−a)+fy(y−b)z=f+fx(x−a)+fy(y−b)
B z=fxx+fyyz=fxx+fyy
C z=fxx(x−a)2z=fxx(x−a)2
D z=fyy(y−b)2z=fyy(y−b)2
The tangent plane is the best 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). It uses first partial derivatives at the point.
A normal vector to tangent plane of z=f(x,y)z=f(x,y) can be
A ⟨fx,fy,0⟩⟨fx,fy,0⟩
B ⟨1,0,0⟩⟨1,0,0⟩
C ⟨−fx,−fy,1⟩⟨−fx,−fy,1⟩
D ⟨0,1,0⟩⟨0,1,0⟩
Writing tangent plane as −fx(x−a)−fy(y−b)+(z−f(a,b))=0−fx(x−a)−fy(y−b)+(z−f(a,b))=0 shows coefficients give a normal direction. Thus ⟨−fx,−fy,1⟩⟨−fx,−fy,1⟩ is normal.
If fxyfxy exists but is not continuous, Clairaut theorem
A Always holds
B May fail
C Gives continuity
D Gives homogeneity
Clairaut’s equality fxy=fyxfxy=fyx is guaranteed when mixed partials are continuous near the point. Without continuity, equality might still hold, but it is not guaranteed.
Total derivative of f(x,y)f(x,y) when x(t),y(t)x(t),y(t) is
A df/dt=fx+fydf/dt=fx+fy
B df/dt=x′+y′df/dt=x′+y′
C df/dt=fxydf/dt=fxy
D df/dt=fxx′+fyy′df/dt=fxx′+fyy′
By chain rule, changes in ff come from both variables: df/dt=∂f/∂x⋅dx/dt+∂f/∂y⋅dy/dtdf/dt=∂f/∂x⋅dx/dt+∂f/∂y⋅dy/dt. This is the total rate of change.
If F(x,y)=0F(x,y)=0 and Fy≠0Fy=0, then dy/dxdy/dx equals
A critical point occurs where first partial derivatives both vanish or fail to exist. ∇f=0∇f=0 means fx=fy=0fx=fy=0, making it a candidate for max, min, or saddle.
If f(tx,ty)=tnf(x,y)f(tx,ty)=tnf(x,y), then ff is
A Periodic in tt
B Always linear
C Not homogeneous
D Homogeneous degree nn
This scaling rule is the definition of homogeneity. The exponent nn tells how output scales when all inputs scale equally, which is central in Euler’s theorem.
For homogeneous f(x,y)f(x,y) of degree nn, Euler identity is
A fx+fy=nffx+fy=nf
B fxx+fyy=nffxx+fyy=nf
C xfx+yfy=nfxfx+yfy=nf
D x+y=nfx+y=nf
Euler’s theorem links the function to its first partial derivatives. It helps verify homogeneity and often simplifies expressions involving fxfx and fyfy.
If f(x,y)f(x,y) is homogeneous of degree 0, then
A xfx+yfy=fxfx+yfy=f
B xfx+yfy=0xfx+yfy=0
C xfx+yfy=2fxfx+yfy=2f
D xfx+yfy=1xfx+yfy=1
Substitute n=0n=0 in Euler’s theorem. Degree 0 means scaling inputs does not change output. The identity becomes xfx+yfy=0xfx+yfy=0, common in ratio-type functions.
Degree of homogeneity of f(x,y)=x2+xyf(x,y)=x2+xy is
A 1
B 0
C 3
D 2
Both terms x2x2 and xyxy scale as t2t2 when (x,y)→(tx,ty)(x,y)→(tx,ty). So f(tx,ty)=t2f(x,y)f(tx,ty)=t2f(x,y), meaning the function is homogeneous of degree 2.
A quick check for homogeneity is to
A Compute Jacobian
B Compute Hessian
C Replace by tx,tytx,ty
D Integrate function
Substituting scaled variables shows whether a single power of tt factors out. If it does, the function is homogeneous and that power is the degree.
The Jacobian ∂(u,v)∂(x,y)∂(x,y)∂(u,v) represents
A Global maximum value
B Local area scaling
C Exact arc length
D Limit existence
Jacobian determinant measures how a small area element in (x,y)(x,y) transforms under mapping to (u,v)(u,v). It gives scaling factor (and orientation sign) locally.
If ∂(u,v)∂(x,y)=0∂(x,y)∂(u,v)=0 at a point, mapping may be
A Always invertible
B Always continuous
C Always linear
D Not locally invertible
A zero Jacobian means local area scaling collapses to zero. The mapping can “fold” or “flatten,” and local one-to-one behavior can fail, so invertibility may not hold.
For inverse transformations, correct relation is
A Juv/xyJxy/uv=1Juv/xyJxy/uv=1
B Juv/xy+Jxy/uv=1Juv/xy+Jxy/uv=1
C Juv/xy=Jxy/uvJuv/xy=Jxy/uv
D Juv/xyJxy/uv=0Juv/xyJxy/uv=0
When maps are inverse and differentiable, one Jacobian is the reciprocal of the other. Their product equals 1, similar to derivative–inverse derivative relations in single-variable calculus.
In polar coordinates, dAdA equals
A dr dθdrdθ
B r2dr dθr2drdθ
C r dr dθrdrdθ
D sinθ dr dθsinθdrdθ
The Jacobian ∣∂(x,y)/∂(r,θ)∣=r∣∂(x,y)/∂(r,θ)∣=r. Hence the area element transforms from dx dydxdy to r dr dθrdrdθ in polar coordinates.
For (u,v)=(x+y,x−y)(u,v)=(x+y,x−y), the Jacobian equals
For composite mappings, Jacobians multiply: ∂(u,v)/∂(x,y)=∂(u,v)/∂(r,s)⋅∂(r,s)/∂(x,y)∂(u,v)/∂(x,y)=∂(u,v)/∂(r,s)⋅∂(r,s)/∂(x,y). This is the multivariable version of chain rule.
A 3-variable Jacobian ∂(u,v,w)∂(x,y,z)∂(x,y,z)∂(u,v,w) is a
A 2×2 determinant
B 1×1 determinant
C 3×3 determinant
D 4×4 determinant
It is the determinant of the matrix of first partial derivatives ∂(u,v,w)/∂(x,y,z)∂(u,v,w)/∂(x,y,z). It measures local volume scaling under the transformation.
In cylindrical coordinates, the Jacobian factor is
A r2r2
B rr
C sinθsinθ
D cosθcosθ
Cylindrical coordinates extend polar with zz. The volume element becomes dV=r dr dθ dzdV=rdrdθdz. The factor rr comes from polar conversion in the xyxy-plane.
In spherical coordinates, common volume element is
A ρ2sinϕ dρ dϕ dθρ2sinϕdρdϕdθ
B ρsinθ dρ dϕ dθρsinθdρdϕdθ
C ρ3 dρ dϕ dθρ3dρdϕdθ
D sinϕ dρ dϕ dθsinϕdρdϕdθ
Standard spherical conversion yields Jacobian factor ρ2sinϕρ2sinϕ. This accounts for radial stretching (ρ2ρ2) and angular stretching (sinϕsinϕ) in 3D space.
A Jacobian becomes negative mainly indicating
A Discontinuity
B Nonexistence of limit
C Differentiability fails
D Orientation reversal
The sign of Jacobian shows whether the mapping preserves or reverses orientation. Magnitude gives scaling. Negative sign does not mean discontinuity; it indicates a “flip” locally.
For f(x,y)f(x,y), second derivative test uses discriminant
A D=fxfyD=fxfy
B D=fxx+fyyD=fxx+fyy
C D=fxxfyy−(fxy)2D=fxxfyy−(fxy)2
D D=fxy+fyxD=fxy+fyx
At a critical point, DD summarizes local curvature. With D>0D>0 we may get min or max depending on fxxfxx. With D<0D<0 we get saddle.
If D>0D>0 and fxx>0fxx>0, the point is
A Local maximum
B Local minimum
C Saddle point
D No conclusion
Positive DD indicates same-type curvature. If fxx>0fxx>0, curvature opens upward like a bowl, giving a local minimum at the critical point.
If D>0D>0 and fxx<0fxx<0, the point is
A Local minimum
B Saddle point
C No conclusion
D Local maximum
D>0D>0 shows consistent curvature. If fxx<0fxx<0, the surface bends downward near the point, forming a local maximum.
If D<0D<0 at a critical point, the point is
A Saddle point
B Local minimum
C Local maximum
D Absolute minimum
Negative DD indicates curvature changes sign in different directions, so the surface rises one way and falls another. That is exactly the saddle point behavior.
If D=0D=0, second derivative test is
A Always minimum
B Always maximum
C Inconclusive
D Always saddle
When D=0D=0, quadratic terms do not decide the nature of the point. You must examine higher-order terms or analyze the function directly near the point.
Hessian matrix for f(x,y)f(x,y) is
A (fxfyfyfx)(fxfyfyfx)
B (fxxfxyfyxfyy)(fxxfyxfxyfyy)
C (f00f)(f00f)
D (xyyx)(xyyx)
The Hessian collects second partial derivatives and captures curvature. It is key in the second derivative test and in multivariable optimization methods.
For constrained extrema with g(x,y)=cg(x,y)=c, Lagrange condition is
A ∇f=∇g∇f=∇g
B ∇f=0∇f=0 always
C f=gf=g
D ∇f=λ∇g∇f=λ∇g
At optimum points on a constraint curve, gradients are parallel because movement along the constraint cannot increase ff first-order. The multiplier λλ captures proportionality.
Total differential approximation for small changes is
A Δf≈fxxΔx2Δf≈fxxΔx2
B Δf≈fyyΔy2Δf≈fyyΔy2
C Δf≈fxΔx+fyΔyΔf≈fxΔx+fyΔy
D Δf≈ΔxΔyΔf≈ΔxΔy
For small input changes, first-order terms dominate. The total differential provides the best linear estimate and is widely used in measurement error and quick approximations.
A sufficient condition for differentiability at a point is
A Existence of limit only
B Continuous first partials
C Existence of partials only
D Equality of mixed partials only
If fxfx and fyfy exist in a neighborhood and are continuous near the point, then ff is differentiable there. This is a practical and commonly used condition.
A tangent direction to level curve f(x,y)=cf(x,y)=c at a point must satisfy
A ∇f×t=0∇f×t=0
B ∇f=t∇f=t
C ∇f+t=0∇f+t=0
D ∇f⋅t=0∇f⋅t=0
Along a level curve, ff stays constant, so the directional derivative along tangent is zero. Since Dtf=∇f⋅tDtf=∇f⋅t, the dot product must be zero.
If ff is continuous at (a,b)(a,b) and f(a,b)≠0f(a,b)=0, then 1/f1/f is
A Continuous there
B Discontinuous there
C Always undefined
D Always constant
Reciprocal of a nonzero continuous function is continuous. The condition f(a,b)≠0f(a,b)=0 ensures 1/f1/f is defined near the point, keeping continuity.
If u=u(x,y)u=u(x,y), v=v(x,y)v=v(x,y), then dudu equals
A u dx+u dyudx+udy
B uxdx+uydyuxdx+uydy
C dx+dydx+dy
D uxdy+uydxuxdy+uydx
The total differential of u(x,y)u(x,y) is du=uxdx+uydydu=uxdx+uydy. It expresses first-order change of uu from small changes in xx and yy.
For f(x,y)=xyf(x,y)=xy, the mixed partial fxyfxy equals
A 0
B xx
C yy
D 1
First fx=yfx=y. Differentiate with respect to yy: fxy=1fxy=1. Similarly fyx=1fyx=1. This is a simple example where mixed partials match.
If ∇f≠0∇f=0 at a point on f(x,y)=cf(x,y)=c, then near that point the level curve is
A Always a circle
B Always a line
C Smooth locally
D Always discontinuous
Nonzero gradient means the level set is not “flat” in all directions. Under standard conditions, it supports a well-defined tangent direction locally, giving a smooth curve behavior.
If f(x,y)f(x,y) is homogeneous of degree nn, then fxfx is homogeneous of degree
A nn
B n−1n−1
C n+1n+1
D 0
Differentiation reduces degree by 1 in homogeneous functions. Scaling (x,y)→(tx,ty)(x,y)→(tx,ty) shows fxfx scales like tn−1tn−1, which is consistent with Euler theorem properties.
If ff is homogeneous of degree nn, then xfx+yfyxfx+yfy is homogeneous of degree
A nn
B n−1n−1
C n+1n+1
D 0
Each term xfxxfx scales as t⋅tn−1=tnt⋅tn−1=tn. Same for yfyyfy. Their sum is homogeneous of degree nn, matching Euler identity xfx+yfy=nfxfx+yfy=nf.
A Jacobian equal to 1 for a transformation suggests
A Local area doubled
B Local area zero
C Limit exists always
D Local area preserved
Jacobian magnitude measures area scaling. If ∣J∣=1∣J∣=1, the mapping preserves local area (though sign may indicate orientation). This is important in coordinate transformations.
For u=x2+y2u=x2+y2, v=tan−1(y/x)v=tan−1(y/x), Jacobian ∂(u,v)∂(x,y)∂(x,y)∂(u,v) is proportional to
A rr
B 1/r1/r
C 1
D r2r2
Here u=r2u=r2 and v=θv=θ. Mapping (x,y)→(r2,θ)(x,y)→(r2,θ) has scaling similar to polar-type relations. The determinant involves rr, reflecting how angular change depends on radius.
Continuity of partial derivatives near a point is mainly used to guarantee
A Limit always exists
B Differentiability there
C Jacobian always zero
D Homogeneity always
If first partial derivatives exist in a neighborhood and are continuous at the point, then the function is differentiable there. This is a common sufficient condition in multivariable calculus.
A saddle point is best described as a point where the function
A Decreases in all directions
B Increases in all directions
C Stays constant nearby
D Increases one way, decreases another
At a saddle point, the surface curves upward along one direction and downward along another. So it is not a local maximum or minimum, even though it may be a critical point.