Chapter 16: Curve Tracing and Polar Coordinates (Set-1)
When a function is concave up on an interval, what is true about f′′(x)f′′(x) there?
A f′′(x)<0f′′(x)<0
B f′′(x)=0f′′(x)=0
C f′′(x)>0f′′(x)>0
D f′(x)=0f′(x)=0
Concave up means the curve bends upward like a cup. In that region, the second derivative is positive, showing the slope f′(x)f′(x) is increasing across the interval.
For a curve to be concave down on an interval, the second derivative should be
A Positive there
B Negative there
C Always zero
D Undefined everywhere
Concave down means the curve bends downward. This happens when f′′(x)<0f′′(x)<0, indicating the slope f′(x)f′(x) decreases as xx increases within that interval.
If the tangent line lies above the curve on an interval, the function is usually
A Concave down
B Concave up
C Constant only
D Periodic only
For concave down functions, the graph lies below its tangents. This geometric property helps in sketching without heavy computation and confirms downward bending behavior.
If the tangent line lies below the curve on an interval, the function is
A Concave down
B Always linear
C Concave up
D Always discontinuous
In concave up regions, the curve stays above its tangent lines. This matches the idea that slopes increase, making the curve bend upward across the interval.
A point where concavity changes sign is called a
A Local maximum
B Vertical asymptote
C Turning point
D Point of inflexion
A point of inflexion occurs when the curve changes from concave up to concave down or vice versa. The key requirement is a real change in concavity.
Which condition is necessary but not sufficient for an inflexion point?
A f′′(a)=0f′′(a)=0
B f′(a)=0f′(a)=0
C f(a)=0f(a)=0
D f′(a)≠0f′(a)=0
Many inflexion points satisfy f′′(a)=0f′′(a)=0, but some points with f′′(a)=0f′′(a)=0 do not change concavity. A sign change of f′′f′′ is the reliable test.
To confirm an inflexion point at x=ax=a, the best simple check is
A f′(a)=0f′(a)=0 only
B f(a)=0f(a)=0 only
C Sign change of f′′f′′
D f′′(a)f′′(a) undefined only
The sure evidence of inflexion is a concavity change. Checking f′′f′′ on both sides of aa and seeing a sign change confirms the graph bends differently.
A stationary inflexion means at that point
A f(a)=0f(a)=0 only
B f′′(a)≠0f′′(a)=0
C Curve is discontinuous
D f′(a)=0f′(a)=0 also
Stationary inflexion is an inflexion point where the tangent is horizontal. So, along with concavity change, the first derivative becomes zero at that point.
A non-stationary inflexion point has
A Nonzero slope there
B Zero function value
C No concavity change
D Only a cusp
Non-stationary inflexion still changes concavity, but the tangent is not horizontal. So f′(a)≠0f′(a)=0, meaning the curve crosses smoothly with a slanting tangent.
Curvature at a point mainly measures
A How sharply it bends
B How sharply it bends
C How fast it moves
D How many roots exist
Curvature quantifies bending of a curve at a point. A larger curvature means a tighter bend, while small curvature means the curve is flatter and closer to a straight line.
The radius of curvature is the reciprocal of
A Slope dy/dxdy/dx
B Second derivative
C Curvature κκ
D Function value
Radius of curvature ρρ is defined as ρ=1/κρ=1/κ. Big radius means gentle bending, while small radius means sharp turning of the curve.
For y=f(x)y=f(x), curvature involves
A y′y′ and y′′y′′
B Only y′y′
C Only y′′y′′
D Only yy
Curvature depends on both slope and how the slope changes. For Cartesian curves, the standard formula uses y′y′ and y′′y′′, capturing direction and bending rate together.
Curvature of a straight line is
A One
B Infinite
C Negative
D Zero
A straight line has no bending, so curvature is zero everywhere. Its radius of curvature is effectively infinite, meaning the curve never turns.
Curvature of a circle of radius RR is
A RR
B R2R2
C 1/R1/R
D 00
A circle bends uniformly. Its curvature stays constant at 1/R1/R. Larger circles bend less, so curvature decreases as radius increases.
A vertical asymptote occurs typically where
A Numerator becomes zero
B Denominator becomes zero
C Derivative becomes zero
D Function becomes constant
For many rational functions, vertical asymptotes appear at values of xx making the denominator zero (after simplification). The function values grow without bound near those xx.
A horizontal asymptote y=Ly=L is found using
A limx→0f(x)limx→0f(x)
B limx→af′(x)limx→af′(x)
C limx→∞f(x)limx→∞f(x)
D limx→af′′(x)limx→af′′(x)
Horizontal asymptotes describe end behavior. If limx→∞f(x)=Llimx→∞f(x)=L (or as x→−∞x→−∞), the graph approaches the line y=Ly=L far away.
For a rational function with same degrees top and bottom, horizontal asymptote is
A Ratio of leading coefficients
B Sum of constants
C Always y=0y=0
D Always y=1y=1
When degrees match, the highest power terms dominate. Their coefficient ratio gives the limit as x→∞x→∞, so the horizontal asymptote is that constant ratio.
If degree of numerator is less than denominator, horizontal asymptote is
A y=1y=1
B x=0x=0
C y=0y=0
D y=xy=x
When the denominator grows faster than numerator, the fraction tends to zero for large ∣x∣∣x∣. Hence the graph approaches the x-axis, giving horizontal asymptote y=0y=0.
A slant asymptote is most common when numerator degree is
A Same as denominator
B Two less than denominator
C One more than denominator
D Much smaller always
When the numerator degree exceeds the denominator by exactly one, long division produces a linear quotient. That line gives the oblique (slant) asymptote for large ∣x∣∣x∣.
To find a slant asymptote of a rational function, we mainly use
A Factor theorem
B Long division
C L’Hospital only
D Binomial theorem
Polynomial long division rewrites the function as “line + remainder/(denominator)”. The remainder part vanishes at infinity, leaving the quotient line as the slant asymptote.
A singular point on an implicit curve generally means
A Function becomes periodic
B Graph is always smooth
C Derivatives become undefined
D Asymptote appears
At a singular point, the curve may not have a unique tangent or smooth behavior. Often dy/dxdy/dx fails or multiple tangents exist, indicating special structure like cusps or nodes.
A cusp is best described as
A Sharp pointed turn
B Smooth max point
C Horizontal asymptote
D Parallel tangents
A cusp is a singular point where the curve meets with a pointed shape and tangent direction changes abruptly. The curve is not smooth there, unlike normal turning points.
A node (double point) usually represents
A End behavior line
B Constant slope region
C Self-intersection point
D One-sided limit only
A node is a double point where the curve crosses itself. Typically two distinct tangents pass through the same point, indicating two branches intersect there.
At a typical double point, you may have
A No tangents
B Exactly one tangent
C Only vertical tangent
D Two tangents
Double points often have two branches meeting, so the tangent equation gives two possible directions. This is a key step in tracing implicit algebraic curves.
In polar coordinates, the point is represented as
A (x,y)(x,y)
B (r,θ)(r,θ)
C (m,c)(m,c)
D (a,b)(a,b)
Polar form uses radius rr from the pole and angle θθ from the initial line. It is especially useful for curves with rotational symmetry and circular patterns.
The pole in polar coordinates corresponds to
A x-intercept only
B y-intercept only
C Origin
D Infinity point
The pole is the fixed reference point from which distance rr is measured. In Cartesian terms, it corresponds to the origin (0,0)(0,0).
The initial line usually corresponds to
A Positive x-axis
B Positive y-axis
C Line y=xy=x
D Any random line
By standard convention, θθ is measured from the positive x-axis. This reference line is called the initial line, helping keep angles consistent.
Conversion from polar to Cartesian is
A x=rsinθ,y=rcosθx=rsinθ,y=rcosθ
B x=θcosrx=θcosr
C y=θsinry=θsinr
D x=rcosθ,y=rsinθx=rcosθ,y=rsinθ
Polar-to-Cartesian conversion uses basic trigonometry. The horizontal component of rr is rcosθrcosθ and the vertical component is rsinθrsinθ.
Conversion from Cartesian to polar uses
A r=x+yr=x+y
B r=xyr=yx
C r=x2+y2r=x2+y2
D r=x2+y2r=x2+y2
The radius rr is the distance from the origin to (x,y)(x,y). Distance formula gives r=x2+y2r=x2+y2, forming the base for converting to polar form.
A common formula for θθ in Cartesian to polar is
A tanθ=y/xtanθ=y/x
B tanθ=x/ytanθ=x/y
C θ=x+yθ=x+y
D θ=xyθ=xy
The angle θθ is measured from the positive x-axis, so tanθtanθ equals opposite/adjacent = y/xy/x. Quadrant awareness is important for correct θθ.
If a polar curve satisfies r( −θ)=r(θ)r(−θ)=r(θ), it is symmetric about
A Pole only
B Line θ=π/2θ=π/2
C Initial line
D No symmetry
Replacing θθ by −θ−θ reflects points across the initial line (x-axis). If the equation stays unchanged, the polar curve has symmetry about that line.
If a polar curve satisfies r(π−θ)=r(θ)r(π−θ)=r(θ), it is symmetric about
A Initial line
B Pole only
C x-axis only
D Line θ=π/2θ=π/2
Substituting π−θπ−θ reflects points about the vertical line through the pole. If the equation remains the same, the curve is symmetric about θ=π/2θ=π/2.
If a polar equation is unchanged when rr is replaced by −r−r, symmetry is about
A Pole
B Initial line
C Vertical axis
D No axis
Replacing rr by −r−r maps a point to the opposite direction through the pole. If the curve is unchanged, it indicates symmetry about the pole (origin symmetry).
A polar curve passes through the pole when
A θ=0θ=0 always
B r≠0r=0 always
C r=0r=0 for some θθ
D r=θr=θ only
The pole is the origin in polar form, so reaching the pole means the radius becomes zero. Solving r(θ)=0r(θ)=0 gives angles where the curve crosses the pole.
In curve tracing, intercepts usually mean points where the curve meets
A Tangents only
B Normals only
C Asymptotes only
D Axes
Intercepts are points where the curve crosses coordinate axes. They help anchor the sketch, showing key locations like x-intercepts (where y=0y=0) and y-intercepts (where x=0x=0).
A basic step to locate turning points for y=f(x)y=f(x) is solving
A f(x)=0f(x)=0
B f′(x)=0f′(x)=0
C f′′(x)=0f′′(x)=0
D f′′′(x)=0f′′′(x)=0
Turning points occur where the slope becomes zero, giving horizontal tangents. Solving f′(x)=0f′(x)=0 provides candidates, then sign change of f′f′ confirms maxima or minima.
The second derivative test for local maximum typically needs
A f′(a)=0,f′′(a)>0f′(a)=0,f′′(a)>0
B f(a)=0,f′(a)=0f(a)=0,f′(a)=0
C f′(a)=0,f′′(a)<0f′(a)=0,f′′(a)<0
D f′′(a)=0f′′(a)=0 only
A local maximum happens when the slope is zero and the curve is concave down at that point. f′′(a)<0f′′(a)<0 shows downward bending, confirming a peak.
The second derivative test for local minimum typically needs
A f′(a)=0,f′′(a)>0f′(a)=0,f′′(a)>0
B f′(a)=0,f′′(a)<0f′(a)=0,f′′(a)<0
C f(a)=0,f′′(a)=0f(a)=0,f′′(a)=0
D f′(a)≠0f′(a)=0
A local minimum occurs where the tangent is horizontal and the curve bends upward. f′′(a)>0f′′(a)>0 indicates concave up behavior, confirming a valley point.
An oblique asymptote is another name for
A Vertical asymptote
B Horizontal asymptote
C Curvature circle
D Slant asymptote
An oblique asymptote is a straight line with nonzero slope approached by the curve at infinity. It typically appears in rational functions when degrees differ by one.
Asymptotes mainly describe curve behavior
A Only at origin
B Only at maxima
C Near infinity or blow-up
D Only at minima
Asymptotes show how a curve behaves far away or near undefined points. They guide long-range sketching and help predict whether the graph approaches certain lines.
Curvature is usually denoted by
A θθ
B κκ
C ρρ
D ππ
The symbol κκ commonly represents curvature. It measures bending per unit arc length. Radius of curvature is denoted by ρρ and equals 1/κ1/κ.
The circle that “best fits” the curve at a point is the
A Unit circle
B Euler circle
C Reference circle
D Osculating circle
The osculating circle matches the curve’s position, tangent direction, and curvature at a point. It provides a local circular approximation and helps visualize radius and center of curvature.
The center of curvature lies on the
A Normal line
B Tangent line
C x-axis always
D y-axis always
The center of the osculating circle is located along the normal direction from the point on the curve. This is because the radius of the circle is perpendicular to the tangent.
The locus of centers of curvature is called
A Involute
B Ellipse
C Evolute
D Parabola
The evolute is formed by tracing all centers of curvature of the original curve. It is closely related to normals and curvature changes and appears in advanced curve-tracing ideas.
A parametric curve is commonly written as
A y=f(x)y=f(x) only
B x=f(t),y=g(t)x=f(t),y=g(t)
C x=g(y)x=g(y) only
D r=f(x)r=f(x) only
In parametric form, both xx and yy depend on a parameter tt. This is useful for curves with loops, self-intersections, and motion-based interpretation.
A quick symmetry test for Cartesian curves is checking invariance under
A y→−yy→−y
B x→x+1x→x+1
C y→y+1y→y+1
D x→2xx→2x
If replacing yy by −y−y leaves the equation unchanged, the curve is symmetric about the x-axis. Similar substitutions test symmetry about y-axis or origin.
For implicit curves, a tangent in slope form often starts from
A Integrating twice
B Factoring only
C Completing square
D Differentiating implicitly
When xx and yy are mixed, implicit differentiation gives dy/dxdy/dx. Using that slope at a point, you can write the tangent line, crucial for curve tracing.
In curve tracing, interval of convexity usually means
A Concave down interval
B Constant function interval
C Concave up interval
D Discontinuous interval
In many texts, “convex” corresponds to concave up behavior. On such intervals, the second derivative is positive and the curve lies above its tangents.
A point with f′′(a)=0f′′(a)=0 but no sign change in f′′f′′ is
A Not an inflexion
B Always an inflexion
C Always a cusp
D Always a node
f′′(a)=0f′′(a)=0 alone cannot guarantee inflexion. If concavity stays the same on both sides, the bending does not change, so it is not a true inflexion point.
A practical curve-tracing checklist usually begins with
A Curvature circle only
B Evolute drawing first
C Domain and intercepts
D Polar length first
Starting with domain, symmetry, and intercepts gives quick structure to the sketch. After that, compute derivatives for monotonicity and concavity, then add asymptotes and special points for accuracy.