Chapter 16: Curve Tracing and Polar Coordinates (Set-2)
While sketching y=f(x)y=f(x), the interval where f′′(x)>0f′′(x)>0 is mainly used to show
A Downward bending
B No curve change
C Upward bending
D Only intercepts
Explanation: When f′′(x)>0f′′(x)>0, the slope is increasing, so the graph bends upward (concave up). This helps decide the “cup” shape while tracing the curve smoothly.
If f′′(x)<0f′′(x)>0 on an interval, the graph of f(x)f(x) is best described as
A Bends downward
B Bends upward
C Perfectly straight
D Always constant
Explanation: A negative second derivative means the slope is decreasing as xx increases. So the curve bends downward (concave down), which is useful in curve tracing and sketching.
A concave function (in basic sketching sense) is commonly one where the curve is
A Above tangents
B Always horizontal
C Always vertical
D Below tangents
Explanation: In common curve-sketching language, “concave” often refers to concave down behavior, where the graph lies below its tangent lines across that interval.
A convex function (in basic sketching sense) is commonly one where the curve is
A Below tangents
B Always linear
C Above tangents
D Always discontinuous
Explanation: A convex (concave up) curve stays above its tangents. This geometric view matches f′′(x)>0f′′(x)>0 and helps sketch the curve without relying only on calculations.
Which statement is correct about an inflexion point?
A Slope must be zero
B Concavity must change
C Function must be zero
D Derivative must fail
Explanation: A true inflexion point is defined by a change of concavity. The slope may be zero or nonzero, but without concavity change, it is not an inflexion.
At an inflexion point x=ax=a, which is most often checked first?
A f(a)=0f(a)=0
B f′(a)≠0f′(a)=0
C f′′(a)=0f′′(a)=0
D f(a)f(a) maximum
Explanation: Many candidates come from solving f′′(a)=0f′′(a)=0 or where f′′f′′ is undefined. Then you verify by checking whether concavity actually changes on either side.
If f′′(a)=0f′′(a)=0 and f′′f′′ changes from negative to positive near aa, then aa is
A Inflexion point
B Local maximum
C Vertical asymptote
D End point only
Explanation: Negative to positive means the curve changes from concave down to concave up. That change confirms an inflexion point, a key feature in accurate curve tracing.
Curvature being large at a point means the curve there is
A Almost straight
B Always horizontal
C Always symmetric
D Very sharply bent
Explanation: Curvature measures how quickly direction changes. Large curvature indicates a tight bend (small radius of curvature), helping identify sharp turns when analyzing paths and sketches.
If curvature κκ is small, the radius of curvature ρρ is
A Very large
B Very small
C Always zero
D Always one
Explanation: Since ρ=1/κρ=1/κ, a small curvature implies a large radius of curvature. This means the curve locally resembles a big circle, appearing almost straight.
The curvature formula for y=f(x)y=f(x) uses the absolute value mainly to make curvature
A Always decreasing
B Always integer
C Always nonnegative
D Always symmetric
Explanation: Curvature measures bending magnitude, not direction. Using absolute value ensures curvature stays nonnegative, so it represents “how much” the curve bends at that point.
A vertical asymptote x=ax=a is usually confirmed by checking
A limf(x)=0limf(x)=0
B limf(x)=±∞limf(x)=±∞
C f′(a)=0f′(a)=0
D f′′(a)=0f′′(a)=0
Explanation: A vertical asymptote occurs where function values grow without bound near a finite xx. Showing the limit becomes ±∞±∞ near x=ax=a confirms it.
If limx→∞f(x)=5limx→∞f(x)=5, the horizontal asymptote is
A x=5x=5
B y=xy=x
C y=5y=5
D x=0x=0
Explanation: When the limit at infinity equals a constant, the graph approaches that constant value. So the line y=5y=5 becomes the horizontal asymptote for large xx.
For a rational function, oblique asymptote is obtained after
A Polynomial division
B Taking square roots
C Using sine rule
D Completing squares
Explanation: Long division rewrites a rational function as a polynomial plus a remainder fraction. The remainder tends to zero at infinity, leaving the polynomial as the asymptote.
In curve tracing, asymptotes are most helpful to predict behavior
A Only near origin
B Only at maxima
C Only at minima
D For large ∣x∣∣x∣
Explanation: Asymptotes describe end behavior or blow-up behavior. They guide how the curve approaches certain lines when xx becomes very large or near points of discontinuity.
A singular point on an implicit curve often indicates
A Perfect smoothness
B Constant curvature
C Non-smooth behavior
D No real points
Explanation: Singular points occur where the curve may have sharp turns, multiple tangents, or undefined slope. These special points must be identified carefully during curve tracing.
A double point is a point where the curve generally has
A No real branch
B Two intersecting branches
C Only one branch
D Only horizontal branch
Explanation: At a double point, two parts of the curve meet or cross. In many cases, two tangents exist there, showing distinct directions of the two branches.
A point where a curve crosses itself is most commonly called
A Node
B Cusp
C Asymptote
D Vertex
Explanation: A node is a self-intersection type double point. Two branches cross with different tangents. This differs from a cusp, where the curve meets with a sharp point.
A cusp differs from a node mainly because a cusp has
A Two crossings
B A horizontal asymptote
C A sharp point
D A constant slope
Explanation: In a cusp, the curve comes to a pointed tip and is not smooth there. In a node, the curve crosses itself with two distinct tangents, forming an intersection.
In polar coordinates, rr represents the
A Angle from axis
B Slope of tangent
C Area parameter
D Distance from pole
Explanation: In polar form, rr is the radius vector length from the pole (origin) to the point. The angle θθ controls direction from the initial line.
In polar coordinates, θθ represents the
A Distance from pole
B Angle from initial line
C Curve’s curvature
D Tangent length
Explanation: θθ is measured from the initial line (usually positive x-axis). It tells the direction of the radius vector, helping plot polar curves effectively.
If a polar curve is unchanged under θ→θ+πθ→θ+π, it usually shows symmetry about
A Initial line
B y-axis only
C Pole
D No symmetry
Explanation: Adding ππ reverses direction. If the equation remains the same, the curve looks identical after a 180° rotation, indicating origin (pole) symmetry.
A quick polar symmetry test about the initial line checks whether the equation is unchanged under
A θ→−θθ→−θ
B r→r+1r→r+1
C θ→θ/2θ→θ/2
D r→2rr→2r
Explanation: Replacing θθ by −θ−θ reflects points across the initial line. If the polar equation stays unchanged, the curve is symmetric about that line.
The polar equation r=ar=a represents a
A Straight line only
B Parabola always
C Circle centered at pole
D Hyperbola always
Explanation: r=ar=a means every point is at constant distance aa from the pole, which is the definition of a circle centered at the origin in polar coordinates.
The polar equation θ=αθ=α represents a
A Circle through pole
B Line through pole
C Hyperbola branch
D Closed loop only
Explanation: Fixing θθ keeps direction constant while rr varies. That traces a straight line (ray) passing through the pole at angle αα.
A curve “passes through the pole” in polar form when
A θ=0θ=0 occurs
B rr is constant
C θθ constant
D r=0r=0 occurs
Explanation: The pole is reached when the radius becomes zero. Solving r(θ)=0r(θ)=0 gives angles where the curve crosses or touches the pole.
In basic curve tracing, the first derivative is mainly used to find
A Increasing/decreasing intervals
B Asymptotes at infinity
C Nodes and cusps
D Curvature centers
Explanation: The sign of f′(x)f′(x) tells whether the function rises or falls. This identifies monotonic intervals and helps locate turning points, an essential step in sketching.
In basic curve tracing, the second derivative is mainly used to find
A Domain of function
B Intercepts only
C Concavity intervals
D Parametric form
Explanation: The sign of f′′(x)f′′(x) determines concave up or concave down intervals. This shapes the curve between key points and helps confirm inflexion points.
A common mistake in finding inflexion points is assuming f′′(a)=0f′′(a)=0 means
A Always maximum
B Always inflexion
C Always minimum
D Always asymptote
Explanation: f′′(a)=0f′′(a)=0 only gives a candidate. Without sign change, concavity may remain same. So it is not sufficient; verification around the point is necessary.
For a rational function, a hole (removable discontinuity) occurs when
A Factor cancels fully
B Degree is higher
C Limit is infinite
D Curve is symmetric
Explanation: If numerator and denominator share a factor and it cancels, the function may have a removable discontinuity at that xx. The graph has a “hole,” not an asymptote.
Which case gives no horizontal asymptote for a rational function?
A Numerator degree smaller
B Degrees equal
C Limit constant exists
D Numerator degree larger
Explanation: If numerator degree exceeds denominator degree, the function does not approach a constant value. It may have an oblique asymptote (degree difference 1) or higher-degree asymptote.
If a curve is symmetric about the y-axis, the equation often stays unchanged under
A y→−yy→−y
B x→x+1x→x+1
C x→−xx→−x
D y→y+1y→y+1
Explanation: Replacing xx with −x−x mirrors points across the y-axis. If the equation remains unchanged, the curve must be symmetric about the y-axis.
If a curve is symmetric about the origin, the equation often stays unchanged under
A (x,y)→(−x,−y)(x,y)→(−x,−y)
B (x,y)→(x,−y)(x,y)→(x,−y)
C (x,y)→(−x,y)(x,y)→(−x,y)
D (x,y)→(x,y+1)(x,y)→(x,y+1)
Explanation: Origin symmetry means the curve looks the same after a 180° rotation. Substituting (−x,−y)(−x,−y) and seeing no change in the equation confirms origin symmetry.
In polar plotting, choosing a few key θθ values helps mainly to get
A Exact curvature only
B Exact tangents only
C Only vertical asymptotes
D Rough curve shape
Explanation: By computing rr for selected angles, you get anchor points that reveal the overall shape and symmetry. This is the standard practical approach to plot polar graphs.
A tangent in polar form generally depends on
A d2r/dθ2d2r/dθ2
B Only rr
C dr/dθdr/dθ
D Only θθ
Explanation: The slope of a polar curve uses both rr and its rate of change dr/dθdr/dθ. This connects angular change to movement in the plane and helps find tangent direction.
A normal line to a curve at a point is always
A Perpendicular to tangent
B Parallel to tangent
C Same as asymptote
D Same as axis
Explanation: The normal is defined as the line perpendicular to the tangent at the point of contact. It is important in curvature and osculating circle concepts.
The center of the osculating circle lies on the
A Tangent direction
B x-axis only
C Normal direction
D y-axis only
Explanation: The radius of the osculating circle points along the normal, not the tangent. Hence its center lies somewhere on the normal line at that point.
A curve with two distinct tangents at the origin indicates the origin is likely a
A Horizontal asymptote
B Double point
C Inflection always
D Turning point only
Explanation: Two distinct tangents through one point usually mean two branches meet there. This is typical of a double point (like a node) in implicit curve tracing.
A curve that touches and turns back at a sharp point suggests a
A Cusp point
B Node point
C Slant asymptote
D Horizontal asymptote
Explanation: A cusp is a pointed singularity where the curve is not smooth and can “turn” sharply. Unlike a node, it does not show a clean crossing of two branches.
In rational curves, an asymptote is often found by studying
A Only derivative sign
B Only intercepts
C Limits at infinity
D Only symmetry
Explanation: End behavior depends on what happens as x→∞x→∞ or x→−∞x→−∞. Limits or division methods reveal whether the curve approaches a constant line or a slant line.
When degrees are equal in a rational function, the curve near infinity approaches
A Vertical line
B Constant line
C Parabola branch
D Circle arc
Explanation: Equal degrees mean leading terms dominate, producing a constant limit equal to the ratio of leading coefficients. That constant becomes the horizontal asymptote for large ∣x∣∣x∣.
In curve sketching, “interval of concavity” means an interval where the curve keeps
A Same intercept count
B Same asymptote count
C Same domain only
D Same bending type
Explanation: On an interval of concavity, the curve remains either concave up or concave down without switching. This is determined by the sign of the second derivative across that interval.
A non-stationary inflexion point can be recognized because
A Curve has cusp
B Two tangents exist
C Tangent not horizontal
D Asymptote crosses
Explanation: Non-stationary inflexion still changes concavity, but the slope is not zero at that point. So the tangent line is not horizontal, unlike stationary inflexion points.
A simple example type that often has an inflexion point is
A Cubic polynomial
B Constant function
C Straight line
D Circle equation
Explanation: Many cubic functions change concavity once, producing an inflexion point in the middle. This makes cubics a standard example for learning concavity and inflexion in sketching.
A curve approaching a line but never meeting it for large ∣x∣∣x∣ suggests the line is
A Normal line
B Asymptote
C Secant line
D Chord line
Explanation: An asymptote is a line the curve approaches arbitrarily closely as xx becomes large or near a discontinuity. The curve may or may not intersect it, depending on the function.
In parametric curves, the parameter is usually denoted by
A kk
B nn
C pp
D tt
Explanation: Parametric equations typically express xx and yy in terms of a parameter tt. This allows describing curves traced by motion, including loops and self-intersections.
A key advantage of parametric form in curve tracing is handling
A Only straight lines
B Only polynomials
C Loops and crossings
D Only asymptotes
Explanation: Parametric form naturally traces direction and repeated points, making it easier to analyze loops, cusps, and self-intersections that can be difficult in y=f(x)y=f(x) form.
In polar plotting, symmetry about the line θ=π/2θ=π/2 is often tested using
A θ→π−θθ→π−θ
B θ→−θθ→−θ
C r→−rr→−r
D r→r+1r→r+1
Explanation: Replacing θθ by π−θπ−θ reflects points across the vertical line through the pole. If the equation remains unchanged, the curve is symmetric about θ=π/2θ=π/2.
In polar coordinates, the line y=xy=x corresponds to angle
A θ=π/2θ=π/2
B θ=π/4θ=π/4
C θ=π/3θ=π/3
D θ=πθ=π
Explanation: The line y=xy=x makes 45° with the positive x-axis. In radians, 45° equals π/4π/4. So points on that line have direction angle θ=π/4θ=π/4.
A curve with f′(x)>0f′(x)>0 on an interval is
A Decreasing there
B Concave up only
C Increasing there
D Has asymptote
Explanation: Positive first derivative means the function rises as xx increases. This gives monotonic increasing behavior and helps locate where the curve moves upward during tracing.
A good final step after finding intercepts, derivatives, and asymptotes is to
A Sketch with labels
B Ignore special points
C Change coordinate system
D Assume symmetry always
Explanation: After collecting key features—domain, intercepts, monotonicity, concavity, inflexion points, and asymptotes—you combine them into a clean labeled sketch to ensure the curve shape is consistent everywhere.