Chapter 15: Applications of Derivatives and Expansions (Set-2)
In monotonicity testing, the interval points are usually split using
A Discontinuity points only
B Zeros of ff
C Roots of f′f′
D Zeros of f′′f′′
To test increasing/decreasing, we find where f′(x)=0f′(x)=0 or undefined. These points split the number line into intervals. Checking the sign of f′(x)f′(x) on each interval gives monotonicity.
If f′(x)=0f′(x)=0 at x=cx=c and f′(x)>0f′(x)>0 on both sides of cc, then cc is
A Local maximum point
B Flat point only
C Local minimum point
D Discontinuous point
If f′(x)f′(x) remains positive around cc, the function keeps increasing through cc. So there is no turning. The point is stationary (flat tangent) but not a maximum or minimum.
A critical point of f(x)f(x) occurs where
A f′(x)=0f′(x)=0 or undefined
B f(x)=0f(x)=0 only
C f′′(x)=0f′′(x)=0 only
D ff is periodic
Critical points are candidates for extrema. They occur when the derivative becomes zero (horizontal tangent) or does not exist (corner/cusp), provided the function value exists at that point.
For absolute extrema on [a,b][a,b], a function should be
A Differentiable everywhere
B Polynomial only
C Continuous on [a,b][a,b]
D Always increasing
By the Extreme Value Theorem, a continuous function on a closed interval [a,b][a,b] must attain an absolute maximum and minimum somewhere in that interval (possibly at endpoints).
If f′′(x)>0f′′(x)>0 on an interval, the graph is
A Concave down
B Always linear
C Always decreasing
D Concave up
A positive second derivative means the slope is increasing. This produces a “cup-shaped” curve (concave up). Concavity is about how the slope changes, not directly about increasing or decreasing.
If f′′(x)<0f′′(x)<0 on an interval, the graph is
A Concave down
B Concave up
C Always constant
D Always increasing
A negative second derivative means the slope is decreasing. The curve bends like a “cap” (concave down). This helps identify maximum behavior and the bending nature of the graph.
A point of inflection is where the graph
A Stops being continuous
B Becomes horizontal
C Changes concavity
D Crosses x-axis
An inflection point occurs where the curve changes from concave up to concave down, or vice versa. Often f′′(c)=0f′′(c)=0 or undefined, but the key requirement is actual sign change of concavity.
If f′′(c)=0f′′(c)=0 but concavity does not change, then cc is
A Not necessarily inflection
B Always inflection point
C Never on graph
D Always maximum point
f′′(c)=0f′′(c)=0 is only a possible inflection candidate. To confirm an inflection point, concavity must change sign around cc. Without sign change, it is not an inflection point.
For tangent to be parallel to x-axis at x=ax=a, we need
A f(a)=0f(a)=0
B f′′(a)=0f′′(a)=0
C f′(a)=1f′(a)=1
D f′(a)=0f′(a)=0
A line parallel to x-axis has slope 0. Since slope of tangent equals f′(a)f′(a), the tangent is horizontal exactly when f′(a)=0f′(a)=0, making aa a stationary point.
The normal is parallel to y-axis when the tangent is
A Vertical
B Slant
C Horizontal
D Undefined always
A normal is perpendicular to the tangent. If tangent is horizontal, the normal is vertical, hence parallel to the y-axis. This happens when the derivative at that point is zero.
If y=f(x)y=f(x), the length of the subtangent at x=ax=a equals
A f(a)f′(a)f′(a)f(a)
B f′(a)f(a)f(a)f′(a)
C f(a)f′(a)f(a)f′(a)
D f(a)+f′(a)f(a)+f′(a)
For curve y=f(x)y=f(x), subtangent length is ∣ydy/dx∣dy/dxy at the point. Here y=f(a)y=f(a) and dy/dx=f′(a)dy/dx=f′(a), giving ∣f(a)f′(a)∣f′(a)f(a).
The length of the subnormal at x=ax=a equals
A f(a)f′(a)f′(a)f(a)
B f′(a)f(a)f(a)f′(a)
C f(a)+f′(a)f(a)+f′(a)
D f(a)f′(a)f(a)f′(a)
For y=f(x)y=f(x), subnormal length is ∣y⋅dy/dx∣∣y⋅dy/dx∣ at the point. So with y=f(a)y=f(a) and slope f′(a)f′(a), it becomes ∣f(a)f′(a)∣∣f(a)f′(a)∣.
In related rates, differentiating x2+y2=r2x2+y2=r2 with respect to time gives
A 2x+2y=2r2x+2y=2r
B 2xx˙+2yy˙=2rr˙2xx˙+2yy˙=2rr˙
C xx˙+yy˙=rxx˙+yy˙=r
D x+y=rx+y=r
When variables depend on time, differentiate each term using chain rule: ddt(x2)=2xx˙dtd(x2)=2xx˙. Similarly for y2y2 and r2r2. This forms the standard related rates equation.
If y=f(x)y=f(x), the differential approximation for f(x+h)f(x+h) is
A f(x)+h2f′(x)f(x)+h2f′(x)
B f(x)+f′′(x)f(x)+f′′(x)
C f(x)+hf′(x)f(x)+hf′(x)
D f(x)−hf′′(x)f(x)−hf′′(x)
For small hh, f(x+h)≈f(x)+hf′(x)f(x+h)≈f(x)+hf′(x). This is the first-order Taylor expansion and is the basis of linear approximation and error estimation in measurements.
The “first derivative test” for extrema uses the sign change of
A f′(x)f′(x)
B f(x)f(x)
C f′′(x)f′′(x)
D f′′′(x)f′′′(x)
The first derivative test checks whether the derivative changes sign across a critical point. Positive to negative indicates a local maximum, negative to positive indicates a local minimum.
A simple geometric optimization setup usually involves expressing area/volume in terms of
A Many independent variables
B No variables at all
C One variable only
D Only trig ratios
In basic optimization, constraints are used to reduce the objective function (area, volume, cost) into a single-variable function. Then you differentiate, find critical points, and test for maximum/minimum.
Rolle’s theorem can be applied to f(x)f(x) on [a,b][a,b] if
A f(a)=f(b)f(a)=f(b)
B f(a)≠f(b)f(a)=f(b)
C f′(a)=f′(b)f′(a)=f′(b) only
D ff is odd only
Rolle’s theorem requires continuity on [a,b][a,b], differentiability on (a,b)(a,b), and equal endpoint values f(a)=f(b)f(a)=f(b). Then it guarantees at least one point cc where f′(c)=0f′(c)=0.
If a function has a sharp corner inside [a,b][a,b], Rolle’s theorem
A Always applies
B Never needs differentiability
C Gives two solutions
D Cannot be applied
Rolle’s theorem requires differentiability on (a,b)(a,b). A sharp corner means derivative does not exist at that point, so the differentiability condition fails, making Rolle’s theorem not applicable.
If ff is continuous on [a,b][a,b] and differentiable on (a,b)(a,b), then LMVT ensures
A Two equal roots
B Always horizontal tangent
C Some tangent equals secant slope
D Always vertical tangent
LMVT guarantees existence of a point cc where f′(c)=f(b)−f(a)b−af′(c)=b−af(b)−f(a). This means slope of tangent at cc equals slope of the secant line joining (a,f(a))(a,f(a)) and (b,f(b))(b,f(b)).
If f(b)>f(a)f(b)>f(a) and LMVT conditions hold, then there exists cc such that f′(c)f′(c) is
A Negative number
B Positive number
C Exactly zero
D Not defined
Here f(b)−f(a)b−a>0b−af(b)−f(a)>0 (assuming b>ab>a). LMVT gives a point cc where f′(c)f′(c) equals that average slope, so f′(c)f′(c) must be positive.
In CMVT, if g(b)≠g(a)g(b)=g(a), the conclusion gives
A f′(c)g′(c)g′(c)f′(c) equals change ratio
B f′(c)=0f′(c)=0
C f(c)=g(c)f(c)=g(c)
D f′′(c)=g′′(c)f′′(c)=g′′(c)
CMVT states that for suitable ff and gg, there exists cc such that f′(c)g′(c)=f(b)−f(a)g(b)−g(a)g′(c)f′(c)=g(b)−g(a)f(b)−f(a). It generalizes LMVT using two functions.
A common special case of CMVT becomes LMVT when
A g(x)=0g(x)=0
B g(x)=f(x)g(x)=f(x)
C g(x)=xg(x)=x
D g(x)=f′(x)g(x)=f′(x)
If we take g(x)=xg(x)=x, then g′(x)=1g′(x)=1, and CMVT gives f′(c)=f(b)−f(a)b−af′(c)=b−af(b)−f(a), which is exactly the statement of Lagrange’s Mean Value Theorem.
Taylor’s theorem gives f(x)f(x) as polynomial plus
A Random constant term
B Only sine term
C Only cosine term
D Remainder term
Taylor’s theorem represents f(x)f(x) near aa as a Taylor polynomial plus a remainder. The remainder measures the approximation error and depends on higher derivatives at some intermediate point.
In Taylor’s theorem, “about aa” means powers of
A (x−a)(x−a)
B (x+a)(x+a)
C (ax)(ax)
D (a−x2)(a−x2)
Taylor expansion about aa is written in powers of (x−a)(x−a): f(a)+f′(a)(x−a)+⋯f(a)+f′(a)(x−a)+⋯. This centers the approximation at x=ax=a and controls error near that point.
The Taylor polynomial of degree 2 contains terms up to
A (x−a)1(x−a)1
B (x−a)2(x−a)2
C (x−a)3(x−a)3
D (x−a)4(x−a)4
Degree 2 Taylor polynomial includes constant, linear, and quadratic terms: f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2f(a)+f′(a)(x−a)+2!f′′(a)(x−a)2. It gives a better approximation than tangent line near aa.
The Maclaurin expansion of sinxsinx begins with
A x−x3/6x−x3/6
B 1−x2/21−x2/2
C 1+x+x21+x+x2
D x2/2+x4x2/2+x4
sinx=x−x33!+x55!−⋯sinx=x−3!x3+5!x5−⋯. This is derived from derivatives at 0. For small xx, the first term dominates, giving sinx≈xsinx≈x.
The Maclaurin expansion of cosxcosx includes only
A Odd powers only
B Even powers only
C All powers
D No powers
cosx=1−x22!+x44!−⋯cosx=1−2!x2+4!x4−⋯. Because derivatives of cosine at 0 alternate between 0 and ±1, only even powers appear in the series.
The Maclaurin expansion of exex contains
A Only even powers
B Only odd powers
C No constant term
D All powers
ex=1+x+x22!+x33!+⋯ex=1+x+2!x2+3!x3+⋯. Since every derivative of exex equals exex, at 0 all derivatives are 1, creating terms of every power.
Standard approximation for ln(1+x)ln(1+x) for small xx is
A x+x22x+2×2
B 1−x+x221−x+2×2
C x−x22x−2×2
D 1+x+x221+x+2×2
The Maclaurin series is ln(1+x)=x−x22+x33−⋯ln(1+x)=x−2×2+3×3−⋯ for ∣x∣<1∣x∣<1. Truncating after two terms gives a useful basic approximation.
A first-order approximation of (1+x)α(1+x)α for small xx is
A 1+αx1+αx
B 1−αx1−αx
C α+xα+x
D xα+1xα+1
Binomial series gives (1+x)α=1+αx+α(α−1)2×2+⋯(1+x)α=1+αx+2α(α−1)x2+⋯ for ∣x∣<1∣x∣<1. First-order approximation keeps only 1+αx1+αx.
Lagrange remainder term includes factor
A (x−a)n(x−a)n only
B (x+a)n+1(x+a)n+1
C (x−a)n−1(x−a)n−1
D (x−a)n+1(x−a)n+1
Lagrange remainder for degree nn Taylor polynomial is Rn=f(n+1)(c)(n+1)!(x−a)n+1Rn=(n+1)!f(n+1)(c)(x−a)n+1. The power n+1n+1 shows error shrinks fast when xx is near aa.
Cauchy remainder is obtained using an idea similar to
A Pythagoras theorem
B Quadratic formula
C Mean value theorem
D Vector product
Cauchy’s form of remainder is derived using a mean value theorem argument with carefully chosen auxiliary functions. It expresses the remainder using derivatives at intermediate points and helps in bounding errors.
For approximation accuracy, increasing Taylor polynomial degree generally
A Improves approximation
B Worsens approximation
C Makes function periodic
D Makes derivative undefined
Higher degree Taylor polynomials include more terms of the series, capturing more local behavior. Near the expansion point, the remainder becomes smaller (under suitable smoothness), so approximation generally improves.
To estimate error using Lagrange remainder, you need a bound on
A f(a)f(a) only
B ∣f(n+1)∣∣f(n+1)∣
C f′(a)f′(a) only
D ∣f(1)∣∣f(1)∣ only
Error size in Lagrange remainder depends on f(n+1)(c)f(n+1)(c). If you can bound ∣f(n+1)(x)∣∣f(n+1)(x)∣ on the interval between aa and xx, you can bound the remainder magnitude.
If you approximate sinxsinx by xx, the neglected next term is about
A x2/2×2/2
B x4/24×4/24
C x5/120×5/120
D x3/6×3/6
sinx=x−x36+⋯sinx=x−6×3+⋯. If you keep only xx, the next omitted term is −x3/6−x3/6. Its size gives a quick estimate of approximation error for small xx.
In curve tracing, a local maximum can occur only at
A Critical points
B f(x)=0f(x)=0 points
C Inflection points only
D Endpoints never
Local extrema can occur where f′(x)=0f′(x)=0 or undefined (critical points). While endpoints matter for absolute extrema on intervals, local maxima/minima in open intervals occur at critical points.
If f′(x)=0f′(x)=0 and f′′(x)>0f′′(x)>0 at x=cx=c, the curve near cc looks like
A Upside cap shape
B Straight vertical line
C Cup shape
D Broken corner
f′′(c)>0f′′(c)>0 means concave up. With f′(c)=0f′(c)=0 giving horizontal tangent, the curve bends upward like a cup, so cc is a local minimum by the second derivative test.
If f′(x)=0f′(x)=0 and f′′(x)<0f′′(x)<0 at x=cx=c, the curve near cc looks like
A Upside cap shape
B Cup shape
C Flat line always
D Discontinuous jump
f′′(c)<0f′′(c)<0 indicates concave down. With a horizontal tangent (f′(c)=0f′(c)=0), the graph forms an upside-down cup, so the point behaves as a local maximum.
For LMVT, equality f(a)=f(b)f(a)=f(b) is
A Necessary always
B Not required
C Same as differentiability
D Same as continuity
LMVT does not need f(a)=f(b)f(a)=f(b). It only needs continuity on [a,b][a,b] and differentiability on (a,b)(a,b). Equal endpoints are required for Rolle’s theorem, which is a special case.
If ff is differentiable on (a,b)(a,b), it must be
A Discontinuous on (a,b)(a,b)
B Constant on (a,b)(a,b)
C Periodic on (a,b)(a,b)
D Continuous on (a,b)(a,b)
Differentiability implies continuity at every point where derivative exists. So if ff is differentiable throughout (a,b)(a,b), it is automatically continuous on (a,b)(a,b).
A basic application of Rolle’s theorem to polynomials is to show that between two roots, there is a root of
A f(x)f(x) again
B f′′(x)f′′(x) only
C f′(x)f′(x)
D f(x)+1f(x)+1
If a polynomial has two distinct roots aa and bb, then f(a)=f(b)=0f(a)=f(b)=0. Rolle’s theorem gives a point c∈(a,b)c∈(a,b) where f′(c)=0f′(c)=0, so derivative has a root between roots.
If a function is continuous on [a,b][a,b] but not differentiable at one interior point, then
A Rolle may fail
B Rolle always works
C LMVT always works
D CMVT never needs g′g′
Rolle’s theorem needs differentiability on (a,b)(a,b). One interior non-differentiable point breaks the condition, so the theorem cannot be applied, even if continuity and endpoint equality still hold.
In Taylor approximation near aa, the best choice of aa is usually
A Far from xx
B Always a=1a=1
C Always a=πa=π
D Close to xx
Taylor polynomial approximates well near the expansion point. Choosing aa near the point where you want the value keeps ∣x−a∣∣x−a∣ small, making higher-power error terms much smaller.
The idea of “error propagation” using differentials mainly uses
A dy=f(x)dxdy=f(x)dx
B dy=f′′(x)dxdy=f′′(x)dx
C dy=f′(x)dxdy=f′(x)dx
D dy=dx/f(x)dy=dx/f(x)
Differentials approximate small changes: Δy≈dy=f′(x) dxΔy≈dy=f′(x)dx. This links measurement error in xx to approximate resulting error in yy, widely used in applied problems.
To choose degree of Taylor polynomial for an accuracy goal, you mainly compare
A Remainder bound
B Endpoints only
C Function symmetry
D Graph color
Degree selection depends on how small you can make the remainder term. Using a bound on ∣f(n+1)∣∣f(n+1)∣ and ∣x−a∣n+1∣x−a∣n+1, you decide the smallest nn giving acceptable error.
A simple limit using series: limx→0ln(1+x)xlimx→0xln(1+x) equals
A 00
B 11
C 22
D Does not exist
ln(1+x)=x−x22+⋯ln(1+x)=x−2×2+⋯. Dividing by xx gives 1−x2+⋯1−2x+⋯, which tends to 1 as x→0x→0. This is a standard Taylor/Maclaurin limit.
Using series: limx→01−cosxx2limx→0x21−cosx equals
A 1/21/2
B 00
C 11
D 22
cosx=1−x22+x424−⋯cosx=1−2!x2+24×4−⋯. So 1−cosx=x22−⋯1−cosx=2×2−⋯. Dividing by x2x2 gives 12−⋯21−⋯, so the limit is 1/21/2.
A “saddle point” in basic intro means a stationary point that is
A Maximum only
B Minimum only
C Not on graph
D Neither max nor min
A saddle point (in this basic sense) has f′(c)=0f′(c)=0 but is not a local maximum or minimum. The function may increase on both sides or show a different turning behavior without extremum.
In Newton–Raphson method for root finding, the iteration uses
A Circle radius
B Area formula
C Tangent intercept
D Matrix inverse
Starting from a guess xnxn, Newton–Raphson uses tangent line at that point to approximate the root. The next guess is xn+1=xn−f(xn)f′(xn)xn+1=xn−f′(xn)f(xn), based on x-intercept.
A common use of derivatives in physics is relating position s(t)s(t) to
A Velocity s′(t)s′(t)
B Area under curve
C Slope of secant only
D Constant acceleration only
If s(t)s(t) is position, then derivative s′(t)s′(t) gives instantaneous velocity. This is a classic application of derivatives as rate of change, connecting calculus directly to motion problems.