Continuity and Differentiability - Practice Questions with Answers

Q1.

A function f is continuous at x = a if:

• A f(a) is defined, without checking the limit
• B lim(x->a) f(x) exists, without comparing it to f(a)
• C lim(x->a) f(x) = f(a)
• D f is differentiable at a, a stronger but different condition

Q2.

Every differentiable function is:

• A Discontinuous
• B Continuous
• C Periodic
• D Unbounded

Q3.

The function f(x) = |x| at x = 0 is:

• A Differentiable but not continuous
• B Continuous and differentiable
• C Continuous but not differentiable
• D Neither continuous nor differentiable

Q4.

d/dx (sin-1 x) =

• A 1/sqrt(1-x²)
• B -1/sqrt(1-x²)
• C 1/(1+x²)
• D 1/sqrt(x²-1)

Q5.

d/dx (cos-1 x) =

• A 1/sqrt(1-x²)
• B -1/sqrt(1-x²)
• C -1/(1+x²)
• D 1/(1+x²)

Q6.

d/dx (tan-1 x) =

• A 1/(1+x²)
• B -1/(1+x²)
• C 1/sqrt(1-x²)
• D 1/(1-x²)

Q7.

A function with LHL not equal to RHL at a point has what kind of discontinuity?

• A Removable
• B Jump discontinuity
• C No discontinuity
• D Infinite discontinuity

Q8.

If y = e^x, then dy/dx =

• A x e^x-1
• B e^x
• C x e^x
• D e

Q9.

Rolle's Theorem requires f(a) and f(b) to satisfy:

• A f(a) > f(b)
• B f(a) = f(b)
• C f(a) < f(b)
• D f(a) times f(b) = 0

Q10.

If x = at² and y = 2at (parametric form), then dy/dx is found using:

• A dy/dx = dx/dt divided by dy/dt
• B dy/dx = (dy/dt) divided by (dx/dt)
• C dy/dx = dy/dt times dx/dt
• D dy/dx = dt/dy

Q11.

Which of these is an example of a removable discontinuity?

• A f(x) = 1/x at x = 0, an infinite discontinuity that grows without bound
• B f(x) = [x] at integer points, a jump discontinuity in the step function
• C f(x) = (x²-1)/(x-1) at x = 1, undefined there but limit exists
• D f(x) = x² everywhere, a polynomial that is already smooth and continuous

Q12.

If f and g are continuous at x = a, then f/g is continuous at a provided:

• A f(a) = 0
• B g(a) = 0
• C g(a) is not equal to 0
• D f(a) is not equal to 0

Q13.

A function f is continuous at x = a if the limit of f(x) as x → a equals:

• A f(a)
• B 0
• C infinity
• D 1

Q14.

A function whose graph can be drawn without lifting the pen is:

• A continuous
• B discontinuous
• C undefined
• D constant only

Q15.

If a function is differentiable at a point, then at that point it is also:

• A continuous
• B discontinuous
• C undefined
• D zero

Q16.

The derivative measures the ___ of a function at a point:

• A instantaneous rate of change
• B the total enclosed area
• C the average value only
• D the solid volume value

Q17.

The function f(x) = |x| fails to be differentiable at x =:

• A 0
• B 1
• C −1
• D 2

Q18.

A jump or break in a graph indicates a point of:

• A discontinuity
• B continuity
• C differentiability
• D symmetry

Q19.

The chain rule is used to differentiate a ___ function:

• A composite
• B constant
• C linear-only
• D polynomial-only

Q20.

The derivative of any constant is:

• A 0
• B 1
• C the constant itself
• D undefined

Q21.

If f(x) = x sin(1/x) for x not 0 and f(0) = 0, is f continuous at x = 0?

• A No, the limit does not exist since sin(1/x) oscillates without settling
• B Yes, since x sin(1/x) is squeezed between -|x| and |x| which go to 0
• C No, because sin(1/x) oscillates infinitely often near x = 0
• D Cannot be determined without evaluating the one-sided limits separately

Q22.

Differentiate y = sin(x² + 1) using the chain rule.

• A cos(x²+1)
• B 2x cos(x²+1)
• C 2x sin(x²+1)
• D cos(2x)

Q23.

If x² + y² = 25, find dy/dx using implicit differentiation.

• A dy/dx = x/y
• B dy/dx = -x/y
• C dy/dx = y/x
• D dy/dx = -y/x

Q24.

Differentiate y = x^x using logarithmic differentiation.

• A dy/dx = x^x
• B dy/dx = x^x (1 + ln x)
• C dy/dx = x times x^x-1
• D dy/dx = x^x ln x

Q25.

If y = tan-1(2x/(1-x²)), which substitution simplifies the derivative using a standard identity?

• A x = sin(theta), reducing the expression to a sine double-angle form
• B x = tan(theta), since 2x/(1-x²) is tan(2theta) when x = tan(theta)
• C x = cos(theta), reducing the expression to a cosine double-angle form
• D x = sec(theta), used for expressions involving square roots of x²-1

Q26.

If x = a cos(t) and y = a sin(t), find dy/dx.

• A tan(t)
• B -tan(t)
• C -cot(t)
• D cot(t)

Q27.

Differentiate y = e^3x cos(2x) using the product rule.

• A e^3x(3cos(2x) - 2sin(2x))
• B e^3x(3cos(2x) + 2sin(2x))
• C 3e^3x cos(2x)
• D e^3x(2cos(2x) - 3sin(2x))

Q28.

For f(x) = |x - 2|, at which point is f not differentiable?

• A x = 0
• B x = 1
• C x = 2
• D f is differentiable everywhere

Q29.

By Rolle's Theorem applied to f(x) = x² - 4x + 3 on [1,3], at which point is f'(c) = 0?

• A c = 1
• B c = 2
• C c = 3
• D c = 1.5

Q30.

Differentiate y = sin-1(x) + cos-1(x).

• A 0
• B 1
• C 2/sqrt(1-x²)
• D pi/2

Q31.

Find dy/dx if y = (sin x)^x.

• A y[x cot x + ln(sin x)]
• B y[x cot x]
• C y[ln(sin x)]
• D x(sin x)^x-1 cos x

Q32.

Differentiate y = ln(x² + 1) with respect to x.

• A 2x/(x²+1)
• B 1/(x²+1)
• C 2/(x²+1)
• D x/(x²+1)

Q33.

Differentiate y = tan-1(x²).

• A 2x/(1+x⁴)
• B 1/(1+x⁴)
• C 2x/(1+x²)
• D x²/(1+x⁴)

Q34.

By the chain rule, the derivative of y = f(g(x)) is:

• A f′(g(x))·g′(x)
• B the product f′(x)·g′(x)
• C the value f(g′(x))
• D the value g′(f(x))

Q35.

The derivative of eˣ with respect to x is:

• A eˣ
• B x·eˣ⁻¹
• C just 1
• D the number e

Q36.

The derivative of ln x with respect to x is:

• A 1/x
• B x
• C ln x
• D eˣ

Q37.

A function that is continuous but not differentiable has a:

• A sharp corner
• B smooth curve
• C flat line
• D single point

Q38.

The derivative of cos x with respect to x is:

• A −sin x
• B sin x
• C cos x
• D −cos x

Q39.

Logarithmic differentiation is used when the variable appears in both the base and the:

• A exponent
• B denominator
• C coefficient
• D constant term

Q40.

The derivative of a product u·v is given by:

• A u′v + uv′
• B u′v′
• C just uv
• D u′v − uv′

Q41.

For what value of k is f(x) = {kx+1 if x<=5, 3x-5 if x>5} continuous at x = 5?

• A k = 1
• B k = 9/5
• C k = 5
• D k = 3

Q42.

If y = tan-1[(sqrt(1+x²) - 1)/x], find dy/dx.

• A 1/(2(1+x²))
• B 1/(1+x²)
• C 2/(1+x²)
• D -1/(2(1+x²))

Q43.

If y = x^{sin x}, find dy/dx.

• A x^{sin x} [cos x ln x + (sin x)/x]
• B x^{sin x} cos x ln x, missing the second additive term
• C x^{sin x} (sin x)/x, missing the logarithmic term
• D sin x times x^{sin x - 1}, a plain power-rule shortcut

Q44.

If x = a(theta - sin theta) and y = a(1 - cos theta), find dy/dx at theta = pi/2.

• A 1
• B 0
• C -1
• D 2

Q45.

For f(x) = x|x|, which statement is correct about differentiability at x = 0?

• A f is not continuous at x=0
• B f is continuous but not differentiable at x=0
• C f is differentiable at x=0 with f'(0)=0
• D f is differentiable at x=0 with f'(0)=1

Q46.

If y = sin-1(2x sqrt(1-x²)) for -1/sqrt(2) <= x <= 1/sqrt(2), find dy/dx.

• A 2/sqrt(1-x²)
• B -2/sqrt(1-x²)
• C 1/sqrt(1-x²)
• D 2/(1-x²)

Q47.

Find d^2y/dx² if y = e^x sin x.

• A 2e^x cos x
• B 2e^x(cos x - sin x)
• C 2e^x sin x
• D e^x(cos x - sin x)

Q48.

If f(x) = |x|³, is f twice differentiable at x = 0?

• A No, f is not even continuous at the origin because the cube of |x| is mistakenly assumed to blow up there
• B No, f is not differentiable at 0 because of the sharp absolute-value kink visible in the graph near zero
• C Yes, f is differentiable but f'' may not exist at 0 in the naive cubic sense, yet checking shows f''(0)=0 exists
• D f is not defined at x=0 since the cube of an absolute value is wrongly treated as an undefined operation

Q49.

Verify Rolle's theorem applicability for f(x) = sin x on [0, pi]. What is f'(c) = 0 satisfied at?

• A c = 0
• B c = pi/2
• C c = pi/4
• D c = pi

Q50.

If y = log(x + sqrt(x²+1)), find dy/dx.

• A 1/sqrt(x²+1)
• B 1/(x+sqrt(x²+1))
• C x/sqrt(x²+1)
• D sqrt(x²+1)/x

Q51.

If x^y = y^x, find dy/dx using implicit and logarithmic differentiation.

• A dy/dx = y(y - x ln y)/(x(x - y ln x))
• B dy/dx = y/x, obtained by treating the exponents as if they cancelled directly
• C dy/dx = x/y, obtained by inverting the y/x ratio without full log differentiation
• D dy/dx = ln(y/x), mistakenly equating the derivative with the log of the ratio

Q52.

For y = xˣ, the derivative dy/dx equals:

• A xˣ(1 + ln x)
• B simply x·xˣ⁻¹
• C simply just xˣ
• D simply ln x

Q53.

The function f(x) = |x| is continuous at x = 0 but not:

• A differentiable
• B well defined
• C real-valued
• D clearly bounded

Q54.

By Rolle’s theorem, if f(a) = f(b) then there is a point c where f′(c) equals:

• A 0
• B 1
• C f(a)
• D b − a

Q55.

The Mean Value Theorem states that f′(c) = [f(b) − f(a)] divided by:

• A (b − a)
• B (a − b)
• C (b + a)
• D ab

Q56.

If y = sin(x²), then dy/dx equals:

• A 2x·cos(x²)
• B cos(x²)
• C 2x·cos x
• D cos(2x)

Q57.

The derivative of tan x with respect to x is:

• A sec²x
• B −cosec²x
• C sec x tan x
• D cot x

Q58.

A function differentiable at every point is necessarily:

• A continuous everywhere
• B plainly discontinuous
• C entirely constant
• D strictly periodic

Q59.

The derivative of the quotient u/v is:

• A (u′v − uv′)/v²
• B (u′v + uv′)/v²
• C u′/v′
• D just uv′

Q60.

For y = e^2x, the derivative dy/dx equals:

• A 2e^2x
• B e^2x
• C 2x·e^2x
• D e^2x/2

Q61.

The function f(x) = |x| is continuous everywhere but not differentiable at:

• A x = 1
• B x = 0
• C x = −1
• D nowhere

Q62.

The value of k that makes f(x) = (sinx)/x for x ≠ 0 and f(0) = k continuous at 0 is:

• A 0
• B 1
• C π
• D undefined

Q63.

If f(x) = x²·sin(1/x) for x ≠ 0 and f(0) = 0, then f'(0) equals:

• A 0
• B 1
• C does not exist
• D ∞

Q64.

The number of points of discontinuity of the greatest integer function [x] on the open interval (0, 3) is:

• A 1
• B 2
• C 3
• D 0

Q65.

If y = x^sinx, then dy/dx equals:

• A x^sinx(cosx·lnx + sinx/x)
• B x^sinx·cosx
• C sinx·x^{sinx − 1}
• D x^sinx·lnx

Q66.

The function f(x) = e^−|x| is continuous everywhere and non-differentiable at:

• A nowhere
• B x = 0
• C x = 1
• D everywhere

Q67.

If f is differentiable with f(x + y) = f(x)f(y), f(0) = 1 and f'(0) = 2, then f'(x) equals:

• A f(x)
• B 2f(x)
• C 2x
• D 2

Q68.

By Rolle's theorem, the value c in (−2, 2) for f(x) = x² − 4 is:

• A −1
• B 0
• C 1
• D 2