Chapter 17: Functions of Several Variables (Set-3)
To prove lim(x,y)→(0,0)f(x,y)=Llim(x,y)→(0,0)f(x,y)=L fails, it is enough to show
A Function is bounded
B One path exists
C Two paths differ
D Partials are zero
A multivariable limit must be the same along every path to the point. If you find just two paths giving two different limiting values, uniqueness breaks and the limit cannot exist.
If f(x,y)→0f(x,y)→0 along every straight line through origin, then
A Limit may still fail
B Limit surely exists
C Continuity is proved
D Differentiability is proved
Checking only lines misses curved paths. Some functions agree on all lines but differ on parabolas or other curves, so the full multivariable limit can still fail.
A strong sufficient method to prove limit =0=0 near origin is to show
A fy(0,0)=0fy(0,0)=0
B fx(0,0)=0fx(0,0)=0
C ∣f(x,y)∣≤g(r)→0∣f(x,y)∣≤g(r)→0
D fxy(0,0)=0fxy(0,0)=0
If you can bound ∣f(x,y)∣∣f(x,y)∣ by a simpler expression g(r)g(r) depending on r=x2+y2r=x2+y2 and g(r)→0g(r)→0, then by squeeze principle the limit is 0.
Using polar form, x2+y2x2+y2 becomes
A rr
B r2r2
C rθrθ
D θ2θ2
With x=rcosθx=rcosθ and y=rsinθy=rsinθ, we get x2+y2=r2(cos2θ+sin2θ)=r2x2+y2=r2(cos2θ+sin2θ)=r2. This simplifies many origin-limit problems.
For f(x,y)=x2yx2+y2f(x,y)=x2+y2x2y at (0,0)(0,0), a good first step is
A Convert to polar
B Use long division
C Find Jacobian
D Use Lagrange
Expressions with x2+y2x2+y2 are ideal for polar conversion. It often turns the function into powers of rr times bounded trig terms, making the limit clear as r→0r→0.
Continuity of f(x,y)f(x,y) at (a,b)(a,b) requires limit equals
A ∇f(a,b)∇f(a,b)
B fx(a,b)fx(a,b)
C fy(a,b)fy(a,b)
D f(a,b)f(a,b)
Continuity means the function value matches the approaching value. So lim(x,y)→(a,b)f(x,y)=f(a,b)lim(x,y)→(a,b)f(x,y)=f(a,b). Derivatives are not part of the definition.
If ff and gg are continuous at (a,b)(a,b), then f+gf+g is
A Discontinuous there
B Differentiable always
C Continuous there
D Constant there
Sums of continuous functions remain continuous. This closure property helps quickly decide continuity of complicated expressions built from known continuous parts.
If ff is continuous and gg is continuous, then f∘gf∘g is
A Continuous (where defined)
B Never continuous
C Only line-continuous
D Continuous only at origin
Composition preserves continuity: if gg is continuous at a point and ff is continuous at gg’s value, then f(g(⋅))f(g(⋅)) is continuous. Domain conditions must still hold.
A typical epsilon–delta statement for continuity at (a,b)(a,b) uses
A Only x-distance
B Distance in R2R2
C Only y-distance
D Only z-distance
In two variables, “closeness” is measured by (x−a)2+(y−b)2(x−a)2+(y−b)2. Continuity means making this distance small enough forces ∣f(x,y)−f(a,b)∣∣f(x,y)−f(a,b)∣ small.
A necessary condition for limit to exist at a point is that
A Partials exist there
B Function is bounded
C All directional limits agree
D Mixed partials equal
If a multivariable limit exists, approaching along any direction must give the same value. Disagreement in any direction immediately shows the limit does not exist.
For f(x,y)f(x,y), the meaning of fx(a,b)fx(a,b) is slope of surface along
A x-axis direction
B y-axis direction
C radial direction
D any direction
fx(a,b)fx(a,b) is computed by varying xx while keeping y=by=b fixed. Geometrically it is the slope of the cross-section curve formed by intersecting the surface with plane y=by=b.
For f(x,y)=x2yf(x,y)=x2y, the partial fyfy is
A 2x2x
B 2xy2xy
C y2y2
D x2x2
Differentiate with respect to yy treating xx constant: ∂(x2y)/∂y=x2∂(x2y)/∂y=x2. This is a standard basic partial derivative rule.
For f(x,y)=x2yf(x,y)=x2y, the partial fxfx is
A x2x2
B 2x2x
C 2xy2xy
D yy
Treat yy as constant. ∂(x2y)/∂x=y⋅2x=2xy∂(x2y)/∂x=y⋅2x=2xy. This shows how the coefficient yy behaves as a constant during differentiation.
If fxyfxy and fyxfyx are continuous near a point, then
A They are opposite
B They are equal
C One is zero
D Both are undefined
Clairaut’s theorem says if second mixed partial derivatives are continuous in a neighborhood, then fxy=fyxfxy=fyx. This lets you swap the order confidently.
For z=f(x,y)z=f(x,y), the linear approximation near (a,b)(a,b) uses
A First partials only
B Second partials only
C Jacobian only
D Euler theorem only
The tangent plane approximation is built from f(a,b)f(a,b), fx(a,b)fx(a,b), and fy(a,b)fy(a,b). It gives the best first-order estimate of ff near the point.
A normal vector to graph z=f(x,y)z=f(x,y) at (a,b)(a,b) can be taken as
A ⟨1,1,1⟩⟨1,1,1⟩
B ⟨fx,fy,0⟩⟨fx,fy,0⟩
C ⟨−fx,−fy,1⟩⟨−fx,−fy,1⟩
D ⟨x,y,z⟩⟨x,y,z⟩
The tangent plane is z−f(a,b)=fx(x−a)+fy(y−b)z−f(a,b)=fx(x−a)+fy(y−b). Writing it as −fxx−fyy+z=constant−fxx−fyy+z=constant shows normal vector ⟨−fx,−fy,1⟩⟨−fx,−fy,1⟩.
Directional derivative is maximum in the direction of
A Normal direction
B Any unit vector
C Tangent direction
D Gradient vector
The directional derivative is ∇f⋅u∇f⋅u. This is largest when uu points in the same direction as ∇f∇f, giving maximum rate of increase.
If ∇f(a,b)=0∇f(a,b)=0, then every directional derivative at (a,b)(a,b) equals
A 0
B 1
C infinity
D depends on path
Directional derivative Duf=∇f⋅uDuf=∇f⋅u. If the gradient is the zero vector, its dot product with any unit vector is zero, so all directional derivatives are zero.
For z=f(x,y)z=f(x,y) with x=x(t),y=y(t)x=x(t),y=y(t), chain rule gives
A dz/dt=fx+fydz/dt=fx+fy
B dz/dt=x′+y′dz/dt=x′+y′
C dz/dt=fxx′+fyy′dz/dt=fxx′+fyy′
D dz/dt=fxydz/dt=fxy
zz changes through both xx and yy. Chain rule adds effects: dz/dt=(∂f/∂x)(dx/dt)+(∂f/∂y)(dy/dt)dz/dt=(∂f/∂x)(dx/dt)+(∂f/∂y)(dy/dt).
For implicit relation F(x,y)=0F(x,y)=0, a basic derivative formula is
A dy/dx=Fx/Fydy/dx=Fx/Fy
B dy/dx=−Fx/Fydy/dx=−Fx/Fy
C dy/dx=Fy/Fxdy/dx=Fy/Fx
D dy/dx=−Fy/Fxdy/dx=−Fy/Fx
Differentiate F(x,y)=0F(x,y)=0: Fx+Fy(dy/dx)=0Fx+Fy(dy/dx)=0. Solving gives dy/dx=−Fx/Fydy/dx=−Fx/Fy when Fy≠0Fy=0. This is the standard implicit differentiation result.
If ff is homogeneous of degree nn, then scaling gives
A f(tx,ty)=f+tf(tx,ty)=f+t
B f(tx,ty)=tff(tx,ty)=tf
C f(tx,ty)=tnff(tx,ty)=tnf
D f(tx,ty)=f/tf(tx,ty)=f/t
Homogeneity means output scales by a power when inputs scale together. The exponent nn is the degree and is checked by substituting (tx,ty)(tx,ty) and factoring tntn.
For degree nn homogeneous f(x,y)f(x,y), Euler theorem states
A xfx+yfy=nfxfx+yfy=nf
B fx+fy=nffx+fy=nf
C fxx+fyy=nffxx+fyy=nf
D x+y=nfx+y=nf
Euler’s theorem links a homogeneous function to its first partial derivatives. It is often used to simplify expressions and to verify computed derivatives quickly.
If f(x,y)=x3+y3f(x,y)=x3+y3, its degree of homogeneity is
A 2
B 1
C 3
D 0
Replace (x,y)(x,y) by (tx,ty)(tx,ty): f(tx,ty)=t3x3+t3y3=t3(x3+y3)=t3f(x,y)f(tx,ty)=t3x3+t3y3=t3(x3+y3)=t3f(x,y). Hence the degree is 3.
If f(x,y)=x2x2+y2f(x,y)=x2+y2x2, the degree is
A 1
B 0
C 2
D −2
Scaling: numerator becomes t2x2t2x2, denominator becomes t2(x2+y2)t2(x2+y2). The t2t2 cancels, so f(tx,ty)=f(x,y)f(tx,ty)=f(x,y). That means degree 0.
Euler theorem in three variables (degree nn) is
A xfx+yfy+zfz=nfxfx+yfy+zfz=nf
B fx+fy+fz=nfx+fy+fz=n
C fxx+fyy+fzz=0fxx+fyy+fzz=0
D x+y+z=nx+y+z=n
The theorem extends by adding a term for each variable. For homogeneous f(x,y,z)f(x,y,z), multiplying each partial by its variable and summing gives nfnf.
∂(u,v)∂(x,y)∂(x,y)∂(u,v) equals
A uxvxuxvx
B ux+vyux+vy
C uy+vxuy+vx
D ∣uxuyvxvy∣uxvxuyvy
The Jacobian is the determinant of the matrix of first partial derivatives. It measures how a small area element in (x,y)(x,y) changes under the mapping to (u,v)(u,v).
If J=∂(u,v)∂(x,y)≠0J=∂(x,y)∂(u,v)=0 at a point, then
A Limit always exists
B Function becomes constant
C Local inverse exists
D Mixed partials equal
A nonzero Jacobian indicates the transformation is locally one-to-one and invertible near that point (basic inverse function principle). Zero Jacobian may indicate collapsing or folding.
For inverse mapping, the correct Jacobian relation is
A Juv/xy Jxy/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/xy Jxy/uv=0Juv/xyJxy/uv=0
When transformations are inverses and derivatives exist properly, the Jacobians are reciprocals. Multiplying them gives 1, just like (du/dx)(dx/du)=1(du/dx)(dx/du)=1 in one variable.
For x=rcosθ,y=rsinθx=rcosθ,y=rsinθ, the Jacobian ∣∂(x,y)∂(r,θ)∣∂(r,θ)∂(x,y) is
A 1/r1/r
B rr
C r2r2
D cosθcosθ
The determinant evaluates to rr. That is why in polar coordinates the area element becomes dA=r dr dθdA=rdrdθ, accounting for larger arc lengths as radius increases.
A “singular” Jacobian at a point means
A Determinant equals one
B Determinant is constant
C Determinant equals zero
D Matrix is diagonal
Singular means the Jacobian matrix loses rank and its determinant vanishes. This indicates the mapping locally collapses area/volume and invertibility can fail at that point.
For cylindrical coordinates, a common conversion is
A x=θr2x=θr2
B x=rsinzx=rsinz
C x=zcosrx=zcosr
D x=rcosθx=rcosθ
Cylindrical coordinates use x=rcosθ,y=rsinθ,z=zx=rcosθ,y=rsinθ,z=z. It extends polar coordinates by adding a height variable zz, useful in 3D problems.
For spherical coordinates, one standard relation is
A x=ρsinϕcosθx=ρsinϕcosθ
B x=ρcosϕcoszx=ρcosϕcosz
C x=rcosθx=rcosθ
D x=ϕθρx=ϕθρ
A common spherical convention is x=ρsinϕcosθx=ρsinϕcosθ, y=ρsinϕsinθy=ρsinϕsinθ, z=ρcosϕz=ρcosϕ. This expresses Cartesian coordinates using radius and angles.
The Jacobian for spherical change of variables is proportional to
A ρsinθρsinθ
B ρ2sinϕρ2sinϕ
C ρ3cosϕρ3cosϕ
D sinθcosϕsinθcosϕ
In standard spherical coordinates, the volume element becomes dV=ρ2sinϕ dρ dϕ dθdV=ρ2sinϕdρdϕdθ. The factor ρ2sinϕρ2sinϕ represents radial and angular stretching.
If u=xu=x, v=yv=y, then ∂(u,v)∂(x,y)∂(x,y)∂(u,v) is
A 0
B −1
C 1
D 2
Here the transformation is identity: ux=1,uy=0,vx=0,vy=1ux=1,uy=0,vx=0,vy=1. Determinant =1⋅1−0⋅0=1=1⋅1−0⋅0=1. Area scaling is unchanged.
A Jacobian used in thermodynamics often represents
A Variable conversion factor
B Current temperature value
C Chemical reaction rate
D Pressure always constant
Thermodynamics uses Jacobians to switch between sets of variables like (P,V,T)(P,V,T). They compactly represent how partial derivatives change under variable transformations and help derive standard identities.
A level surface f(x,y,z)=cf(x,y,z)=c has gradient that is
A Tangent to surface
B Parallel to x-axis
C Normal to surface
D Always zero
Along a level surface, the function value stays constant, so motion tangent to the surface causes no change. The gradient points in greatest increase, hence it must be perpendicular to the surface.
If D=fxxfyy−(fxy)2>0D=fxxfyy−(fxy)2>0 and fxx<0fxx<0, then
A No conclusion
B Local minimum
C Saddle point
D Local maximum
D>0D>0 indicates consistent curvature. If fxx<0fxx<0, the surface bends downward in the xx-direction and similarly nearby, giving a local maximum at the critical point.
If D<0D<0 at a critical point, then
A Saddle point
B Local minimum
C Local maximum
D Global minimum
D<0D<0 means curvature changes sign in different directions. The surface rises in one direction and falls in another, so the point is a saddle, not a local extremum.
If D=0D=0 in second derivative test, then
A Always minimum
B Always maximum
C Need higher analysis
D Always saddle
When D=0D=0, the quadratic approximation is insufficient to classify the point. One must check higher-order terms, use alternative tests, or analyze behavior directly.
Lagrange method for f(x,y)f(x,y) with constraint g(x,y)=cg(x,y)=c uses
A ∇f=∇g∇f=∇g
B ∇f=λ∇g∇f=λ∇g
C ∇f=0∇f=0 always
D f=gf=g always
At constrained extrema, the gradients are parallel. Lagrange multipliers introduce λλ to capture this condition and solve with the constraint equation together.
A Hessian matrix for f(x,y)f(x,y) contains
A First partial derivatives
B Only function values
C Second partial derivatives
D Only Jacobians
The Hessian is the matrix (fxxfxyfyxfyy)(fxxfyxfxyfyy). It summarizes second-order curvature information and is used in classification of critical points.
Error estimation using differential mainly uses
A ∣df∣∣df∣ approximation
B Exact ff values
C Jacobian equals zero
D Euler degree test
For small measurement errors Δx,ΔyΔx,Δy, the change in ff is approximated by df=fxΔx+fyΔydf=fxΔx+fyΔy. Taking absolute values helps estimate maximum possible error.
If fxfx and fyfy exist at a point, continuity is
A Guaranteed always
B Equivalent always
C Never true
D Not guaranteed
Existence of partial derivatives alone does not ensure continuity. A function can have partial derivatives at a point yet still fail to be continuous there because behavior along other paths can misbehave.
If ff is differentiable at a point, then ff is
A Discontinuous there
B Unbounded there
C Continuous there
D Nonexistent there
Differentiability implies a valid linear approximation near the point, which forces the function values to approach the point’s value smoothly. So differentiability always implies continuity.
A common sufficient condition for differentiability is
A Only limit exists
B Continuous first partials
C Only directional limits
D Only mixed partials
If fxfx and fyfy exist in a neighborhood and are continuous near the point, then ff is differentiable there. This is a standard practical differentiability condition.
For f(x,y)f(x,y), the relation between gradient and directional derivative is
A Dot product rule
B Cross product rule
C Product of gradients
D Integral of gradient
Directional derivative in unit direction uu is Duf=∇f⋅uDuf=∇f⋅u. It measures how much of the gradient points along that direction.
A tangent plane gives best approximation of ff near point because it is
A Second-order quadratic
B Global exact surface
C First-order linear model
D Random local fit
The tangent plane uses first derivatives to build the best linear approximation. For small changes, linear terms dominate, so it gives accurate local estimates before higher-order terms matter.
If ∇f∇f is normal to level curve f(x,y)=cf(x,y)=c, then tangent direction satisfies
A ∇f×t=0∇f×t=0
B ∇f+t=0∇f+t=0
C ∇f=t∇f=t
D ∇f⋅t=0∇f⋅t=0
A tangent vector tt lies along the level curve, so moving in that direction does not change ff. Hence the change rate is zero, meaning ∇f⋅t=0∇f⋅t=0.
Implicit function theorem (intro) typically requires at point
A F=1F=1
B Fy≠0Fy=0
C Fx=0Fx=0
D y=0y=0
If F(x,y)=0F(x,y)=0 and Fy≠0Fy=0 at a point, then locally yy can be expressed as a differentiable function of xx. This supports local solvability.
Differentiability is stronger than continuity because it requires
A Linear approximation holds
B Only limit exists
C Only value exists
D Only boundedness
Continuity only needs function values to approach smoothly. Differentiability demands a good linear model with small error compared to distance, ensuring smooth behavior and well-defined rates of change in all directions.