Chapter 16: Curve Tracing and Polar Coordinates (Set-3)
For a twice-differentiable function, the condition “f′(x)f′(x) increasing on an interval” directly implies
A f′′(x)≤0f′′(x)≤0
B f′(x)=0f′(x)=0
C f′′(x)≥0f′′(x)≥0
D f(x)=0f(x)=0
If f′(x)f′(x) is increasing, its rate of change is nonnegative. Since f′′(x)f′′(x) is the derivative of f′(x)f′(x), this means f′′(x)≥0f′′(x)≥0 on that interval.
A curve is concave down on (a,b)(a,b). The slope f′(x)f′(x) on (a,b)(a,b) is
A Increasing function
B Decreasing function
C Always zero
D Not defined
Concave down means f′′(x)<0f′′(x)<0. A negative second derivative implies the first derivative f′(x)f′(x) decreases as xx increases, so the slope becomes smaller across the interval.
If f′′(a)=0f′′(a)=0 and f′′′(a)≠0f′′′(a)=0, then x=ax=a is generally a
A Likely inflexion
B Vertical asymptote
C Local maximum
D Local minimum
When f′′(a)=0f′′(a)=0 and the next derivative is nonzero, the concavity typically changes at aa. This is a common higher-derivative clue for an inflexion point, then confirm by sign change.
A stationary inflexion point must satisfy both
A f(a)=0f(a)=0
B f′′(a)≠0f′′(a)=0
C f′(a)=0f′(a)=0
D f′(a)≠0f′(a)=0
“Stationary” means horizontal tangent, so f′(a)=0f′(a)=0. It is still an inflexion only if concavity changes around aa. The function value need not be zero.
A non-stationary inflexion point is an inflexion point where
A f′(a)=0f′(a)=0
B f(a)=0f(a)=0
C f′′(a)<0f′′(a)<0
D f′(a)≠0f′(a)=0
Non-stationary inflexion means the concavity changes but the tangent is not horizontal. Therefore the slope is not zero at that point, so f′(a)≠0f′(a)=0.
At an inflexion point, which statement is always true?
A Concavity changes
B Slope is zero
C Function is zero
D f′′f′′ undefined
The defining feature of inflexion is change of concavity. f′′(a)=0f′′(a)=0 is common but not guaranteed, and f′(a)f′(a) can be zero or nonzero.
For y=f(x)y=f(x), the radius of curvature ρρ at a point depends on
A Only yy
B Only y′y′
C y′y′, y′′y′′
D Only y′′y′′
Curvature uses both slope and its change. The standard formula combines y′y′ and y′′y′′, then radius of curvature is the reciprocal of curvature.
If a curve is “almost straight” near a point, then curvature κκ near that point is
A Always one
B Always negative
C Very large
D Very small
A nearly straight curve bends very little, so curvature is close to zero. Correspondingly, the radius of curvature becomes very large, matching a large “fitting circle.”
The center of curvature at a point lies on the
A Tangent line
B Normal line
C x-axis only
D y-axis only
The osculating circle has radius along the normal direction, perpendicular to the tangent. Therefore its center must lie on the normal line at that point.
The evolute of a plane curve is the locus of
A Curvature centers
B Tangent points
C Asymptote points
D Turning points
An evolute is formed by tracing the centers of curvature of the original curve. It is closely related to normals and how curvature changes along the curve.
For f(x)=p(x)q(x)f(x)=q(x)p(x), a vertical asymptote at x=ax=a requires
A p(a)=0p(a)=0 always
B p′(a)=0p′(a)=0 always
C q(a)=0q(a)=0 after simplification
D f′′(a)=0f′′(a)=0
First cancel common factors. If the simplified denominator is zero at aa and the function blows up there, x=ax=a is a vertical asymptote, not a removable hole.
A removable discontinuity (“hole”) in a rational function occurs when
A Common factor cancels
B Degrees are equal
C Limit is infinite
D No factors exist
If numerator and denominator share a factor (x−a)(x−a) and it cancels, the original function is undefined at aa but the simplified expression is finite—creating a hole, not an asymptote.
For a rational function with numerator degree exactly one more than denominator degree, the asymptote is usually
A Horizontal line
B Vertical line
C Slant line
D Circular arc
When degree difference is 1, long division yields a linear quotient plus a remainder fraction. The remainder tends to zero at infinity, so the linear quotient becomes the oblique (slant) asymptote.
If degree(numerator) < degree(denominator), then as x→∞x→∞, f(x)f(x) tends to
A 1
B 0
C ∞∞
D −∞−∞
The denominator grows faster, so the ratio goes to zero. Hence y=0y=0 becomes the horizontal asymptote, a key end-behavior result in curve tracing.
If degrees are equal for p(x)q(x)q(x)p(x), the horizontal asymptote is
A Leading coefficient ratio
B Constant term ratio
C Product of degrees
D Sum of roots
For equal degrees, the highest power terms dominate. Their leading coefficients determine the limit at infinity, giving a constant horizontal asymptote equal to that ratio.
A curve intersects its slant asymptote when the difference “curve − asymptote” becomes
A Always positive
B Always infinite
C Zero at some xx
D Undefined always
If f(x)−(mx+c)=0f(x)−(mx+c)=0 for some real xx, the curve meets the asymptote there. Asymptotes can be crossed; they mainly describe behavior as ∣x∣→∞∣x∣→∞.
In implicit curve tracing, a point can be singular if both partial derivatives vanish, meaning
A F=0F=0 only
B Fx=0Fx=0 and Fy=0Fy=0
C Fx≠0Fx=0 always
D Fy≠0Fy=0 always
For F(x,y)=0F(x,y)=0, a regular point typically has at least one of Fx,FyFx,Fy nonzero. If both are zero at a point on the curve, tangency can be non-unique, creating a singularity.
A node is a double point where the curve has
A One real tangent
B No tangent
C Infinite curvature
D Two real tangents
At a node, two branches cross with distinct directions, producing two real tangents. This distinguishes it from a cusp where the curve meets in a sharp point with a different tangency behavior.
A cusp point on a curve is typically identified by
A One repeated tangent
B Two crossings always
C Horizontal asymptote
D Constant curvature
A cusp often has a single tangent direction but the curve approaches it from both sides in a sharp way, giving a repeated tangent rather than two distinct tangents like a node.
A “self-intersection” in curve tracing is most directly linked with a
A Inflexion point
B Horizontal asymptote
C Double point
D Convex interval
Self-intersection means two different parameter values or branches lead to the same point. That is exactly the idea of a double point, frequently classified as a node or related type.
In polar coordinates, converting (r,θ)(r,θ) to Cartesian uses
A x=rsinθx=rsinθ
B x=rcosθx=rcosθ
C x=θcosrx=θcosr
D x=r+θx=r+θ
Polar to Cartesian uses projections of the radius vector on axes: x=rcosθx=rcosθ, y=rsinθy=rsinθ. These conversions are essential for relating polar curves to standard forms.
Which Cartesian expression always equals r2r2 in polar form?
A x+yx+y
B x2−y2x2−y2
C x2+y2x2+y2
D xyxy
Since r=x2+y2r=x2+y2, squaring gives r2=x2+y2r2=x2+y2. This identity helps quickly convert equations between coordinate systems.
A polar equation unchanged under r→−rr→−r (with same θθ) indicates symmetry about
A Pole
B Initial line
C θ=π/2θ=π/2
D x-axis only
Replacing rr by −r−r moves the point to the opposite direction through the pole. If the equation is unchanged, the curve is symmetric under 180° rotation about the pole.
For a polar curve, symmetry about the line θ=π/2θ=π/2 is tested by
A θ→−θθ→−θ
B r→−rr→−r
C r→r+1r→r+1
D θ→π−θθ→π−θ
The substitution θ→π−θθ→π−θ reflects points across the vertical axis through the pole. If the equation stays the same, the curve is symmetric about θ=π/2θ=π/2.
For the polar curve r=2acosθr=2acosθ, the curve is a
A Parabola
B Circle
C Hyperbola
D Spiral
r=2acosθr=2acosθ converts to x2+y2=2axx2+y2=2ax, which is a circle of radius aa centered at (a,0)(a,0). This is a standard result in polar curve tracing.
For the polar curve r=2asinθr=2asinθ, the circle’s center lies on the
A Positive x-axis
B Negative x-axis
C Positive y-axis
D Negative y-axis
r=2asinθr=2asinθ becomes x2+y2=2ayx2+y2=2ay, a circle of radius aa centered at (0,a)(0,a). Recognizing this helps quickly sketch without plotting many points.
If a polar curve crosses the pole, a common additional check is whether rr changes sign near the root to detect
A Loop formation
B Vertical asymptote
C Horizontal asymptote
D Constant curvature
When r=0r=0 occurs, the curve passes the pole. If rr changes sign, the plotted direction reverses, often producing loops or petals, important for correct polar tracing.
For parametric curves x=f(t)x=f(t), y=g(t)y=g(t), the slope dy/dxdy/dx equals
A (dx/dt)/(dy/dt)(dx/dt)/(dy/dt)
B dx/dtdx/dt only
C (dy/dt)/(dx/dt)(dy/dt)/(dx/dt)
D dy/dtdy/dt only
In parametric form, both xx and yy vary with tt. Using chain rule, dy/dx=(dy/dt)/(dx/dt)dy/dx=(dy/dt)/(dx/dt) when dx/dt≠0dx/dt=0, giving tangent slope.
A parametric point is stationary on the curve when
A x=0x=0 only
B dx/dt=0dx/dt=0 and dy/dt=0dy/dt=0
C y=0y=0 only
D t=0t=0 only
If both derivatives vanish, the velocity vector is zero, so motion pauses at that point. Such points can create cusps or special behavior in parametric curve tracing.
In parametric tracing, a loop is suggested when the curve returns to the same point for
A Same tt only
B Only xx repeats
C Only yy repeats
D Two different tt
If distinct parameter values give the same (x,y)(x,y), the curve intersects itself or forms a loop. This is a key idea in identifying double points and self-intersections.
For an implicit curve F(x,y)=0F(x,y)=0, if Fy≠0Fy=0 at a point, the slope dy/dxdy/dx is
A −Fy/Fx−Fy/Fx
B Fx+FyFx+Fy
C −Fx/Fy−Fx/Fy
D Fx−FyFx−Fy
Differentiating F(x,y)=0F(x,y)=0 gives Fx+Fy(dy/dx)=0Fx+Fy(dy/dx)=0. Solving yields dy/dx=−Fx/Fydy/dx=−Fx/Fy (when Fy≠0Fy=0), a standard implicit differentiation result.
A vertical tangent on F(x,y)=0F(x,y)=0 is more likely when
A Fy=0Fy=0, Fx≠0Fx=0
B Fx=0Fx=0, Fy≠0Fy=0
C Fx≠0Fx=0, Fy≠0Fy=0
D Fx=0Fx=0, Fy=0Fy=0 only
From dy/dx=−Fx/Fydy/dx=−Fx/Fy, if Fy=0Fy=0 and Fx≠0Fx=0, the slope becomes infinite, indicating a vertical tangent at that point (unless it is singular).
In curve tracing, checking symmetry first mainly reduces work because it lets you sketch
A All asymptotes only
B Only intercepts
C Only curvature
D One part only
If a curve is symmetric, you can trace only one half or one quadrant and reflect the result. This saves time and helps avoid mistakes in repetitive calculations.
A function can have an inflexion point even if f′(a)≠0f′(a)=0 because
A Maximum must exist
B Minimum must exist
C Concavity can change
D Asymptote appears
Inflexion is about change in concavity, not about slope being zero. A curve may change from concave down to concave up while still having a non-horizontal tangent.
For curve sketching, the “second derivative test” is mainly applied at points where
A f′(x)=0f′(x)=0
B f(x)=0f(x)=0
C f′′(x)=0f′′(x)=0
D f′′′(x)=0f′′′(x)=0
The second derivative test classifies stationary points: if f′(a)=0f′(a)=0 and f′′(a)>0f′′(a)>0, it’s a local minimum; if f′′(a)<0f′′(a)<0, it’s a local maximum.
If f′(a)=0f′(a)=0 and f′′(a)>0f′′(a)>0, the point (a,f(a))(a,f(a)) is a
A Local maximum
B Inflexion always
C Local minimum
D Vertical asymptote
Positive second derivative means concave up, so the stationary point sits at the bottom of a “cup.” This is the standard second derivative test for a local minimum.
If f′(a)=0f′(a)=0 and f′′(a)<0f′′(a)<0, the point (a,f(a))(a,f(a)) is a
A Local minimum
B Node point
C Cusp point
D Local maximum
Negative second derivative means concave down, so the stationary point is at the top of a “cap.” This confirms a local maximum under the second derivative test.
If f′(a)=0f′(a)=0 and f′′(a)=0f′′(a)=0, the second derivative test is
A Always maximum
B Inconclusive
C Always minimum
D Always inflexion
When f′′(a)=0f′′(a)=0, the test cannot classify the point. You must use sign change of f′f′ or higher derivative tests, because the point may be max, min, or neither.
A curve with “tangent horizontal but no max/min” is often a
A Vertical asymptote
B Node point
C Stationary inflexion
D Slant asymptote
A stationary inflexion has f′(a)=0f′(a)=0 but concavity changes, so it is not a local max or min. The curve flattens then continues through with bending switching sides.
For rational functions, “degree comparison method” is mostly used to identify
A End behavior type
B Singular points type
C Curvature at origin
D Double points only
Comparing degrees of numerator and denominator quickly tells whether the asymptote is horizontal, slant, or polynomial of higher degree. This gives fast end-behavior information for curve tracing.
A polar curve with equation unchanged under θ→−θθ→−θ is symmetric about
A Pole only
B θ=π/2θ=π/2
C No symmetry
D Initial line
Replacing θθ by −θ−θ reflects the point across the initial line. If the equation is unchanged, the curve must be symmetric about that initial line.
When converting a polar equation to Cartesian, a common first step is substituting
A x=rsinθx=rsinθ, y=rcosθy=rcosθ
B x=θrx=θr, y=ry=r
C x=rcosθx=rcosθ, y=rsinθy=rsinθ
D x=r+θx=r+θ, y=r−θy=r−θ
These identities connect polar and Cartesian coordinates. Often you also use r2=x2+y2r2=x2+y2 to eliminate rr and θθ and obtain a standard Cartesian equation.
In curve tracing, “tangent and normal in implicit form” is mostly used to study
A Local direction
B Global symmetry
C Only asymptotes
D Only intercepts
Tangent and normal describe how the curve is oriented at a given point. For implicit curves, implicit differentiation gives slope, then the normal is perpendicular to it.
A practical checklist step “behavior near vertical asymptote” means checking
A Only derivatives
B One-sided limits
C Only curvature
D Only symmetry
Near x=ax=a, the curve may go to +∞+∞ on one side and −∞−∞ on the other. One-sided limits show the correct approach direction and shape around the asymptote.
A polar curve’s “initial line” is typically taken as
A Positive y-axis
B Negative x-axis
C Line y=xy=x
D Positive x-axis
By convention, polar angles θθ are measured from the positive x-axis, called the initial line. This standard reference keeps polar graphs consistent.
If a polar curve is unchanged under (r,θ)→(−r,θ+π)(r,θ)→(−r,θ+π), it indicates the same physical point because
A Curve becomes linear
B Curvature becomes zero
C Coordinates are equivalent
D Asymptote vanishes
In polar coordinates, (r,θ)(r,θ) and (−r,θ+π)(−r,θ+π) represent the same point: reversing radius and adding 180° keeps the location unchanged. This helps in symmetry and plotting.
In basic polar tracing, testing symmetry first is useful because it can reduce plotting angles by
A Half or more
B One extra point
C No reduction
D Random reduction
If symmetry exists about initial line, vertical line, or pole, you can plot only a portion of angles and reflect. This often cuts the work roughly in half or even more.
For curvature in motion paths, higher curvature usually suggests
A Straighter motion
B Sharper turn
C Constant speed
D Zero velocity
Curvature measures turning rate of the path. Higher curvature means the path direction changes faster, so the motion follows a tighter bend compared to low curvature paths.
A polar asymptote concept is most relevant when rr becomes unbounded as
A rr approaches zero
B xx approaches zero
C θθ approaches value
D yy approaches zero
Some polar curves blow up for particular angles, meaning r→∞r→∞ as θθ approaches a specific value. That can indicate a polar asymptote direction line.
A good way to avoid missing special points in curve tracing is to check
A Domain and singularities
B Only final sketch
C Only one derivative
D Only symmetry test
Many sketching errors occur by ignoring where the function is undefined or where tangents fail. Checking domain, discontinuities, and singular points early prevents wrong curve branches and missing features.