Continuity and Differentiability: Class 12 Maths Practice Questions

A civil engineer models the vertical profile of a road bridge. The height h(x) = √(4x + 9) metres at horizontal distance x metres (x ≥ 0) from the start of the bridge is used to verify a structural identity involving derivatives.

A civil engineer is modelling the vertical profile of a road bridge. The height (in metres) of the bridge deck above ground at a horizontal distance x metres from the start is given by

h(x) = √(4x + 9), x ≥ 0.

The engineer needs to verify a structural identity involving the rate of change of height and its acceleration (second derivative) to ensure the design is consistent.

(i) Find dh/dx. [1 mark]
(ii) Find d²h/dx². [1 mark]
(iii) Show that h · (d²h/dx²) + (dh/dx)² = 0, and interpret what this identity tells the engineer about the curvature formula for this bridge profile. [2 marks]

A civil engineer models the deflection of a beam using piecewise and implicit functions to study continuity and slope behaviour at critical points along the beam's length.

A civil engineer is analysing the stress distribution along a beam. The deflection function of the beam is modelled as:

f(x) = { (x² sin(1/x) + 3x, x ≠ 0
{ 0, x = 0

(i) The engineer first checks whether the deflection function is continuous at x = 0. Verify whether f(x) is continuous at x = 0. [1 mark]

(ii) Next, the engineer needs to confirm differentiability at x = 0 to ensure no abrupt change in slope. Check whether f(x) is differentiable at x = 0. [1 mark]

(iii) For a different section of the beam, the deflection satisfies the implicit relation:

x² + y² − 2x − 4y + 3 = 0

The engineer needs to find dy/dx at the point (2, 1) to determine the slope of the deflection curve at that point. Find dy/dx at (2, 1).

OR

For yet another section, the deflection is given by y = (cos x)^(sin x), x ∈ (0, π/2). Find dy/dx. [2 marks]

A civil engineer is designing a flyover ramp whose elevation profile is modelled by the function f(x) = { (√(1 + 3x) − 1) / x for x ≠ 0; k for x = 0 }, where x represents horizontal displacement (in metres) from the ramp entry and f(x) represents the normalised height. The ramp must have a smooth, continuous profile with no abrupt changes in elevation.

A civil engineer is designing a flyover ramp whose elevation profile is modelled by the function

f(x) = { (√(1 + 3x) − 1) / x, for x ≠ 0
{ k, for x = 0

where x (in metres) represents the horizontal displacement from the entry point of the ramp and f(x) represents the normalised height.

(i) For the ramp profile to be smooth (no sudden jump), the function f must be continuous at x = 0. Find the value of k that ensures this. [1 mark]

(ii) Once k is set as found in part (i), verify that f is indeed continuous at x = 0 by checking all three conditions. [1 mark]

(iii) The engineer also considers a modified steepness function g(x) = √(log(1 + sin(x²))) for x near the ramp entry. Find dg/dx. [2 marks]

A civil engineer is modelling the vertical cross-section of a road ramp. The height (in metres) at horizontal distance x (in metres) is given by:

f(x) = { 2x² + x, x ≤ 1
{ ax + b, x > 1

For the ramp to be smooth, f must be continuous and differentiable at x = 1.

Definitions provided:
L.H.D.: Lf′(a) = lim[h→0⁻] [f(a−h) − f(a)]/(−h)
R.H.D.: Rf′(a) = lim[h→0⁺] [f(a+h) − f(a)]/h
f is differentiable at x = a if L.H.D. = R.H.D.

A civil engineer is modelling the vertical cross-section of a road ramp that joins a lower road to an elevated highway. The height of the ramp (in metres) at a horizontal distance x (in metres) from the start is modelled by the function

f(x) = { 2x² + x, x ≤ 1
{ ax + b, x > 1

For the ramp profile to be smooth (no abrupt bends or breaks), the height function must be both continuous and differentiable at x = 1. Using the definitions:

Left Hand Derivative (L.H.D.): Lf′(a) = lim[h→0⁻] [f(a − h) − f(a)] / (−h)
Right Hand Derivative (R.H.D.): Rf′(a) = lim[h→0⁺] [f(a + h) − f(a)] / h

Answer the following questions:
(i) Find the value of f(1) using the expression valid at x = 1. [1]
(ii) Find the condition on a and b so that f is continuous at x = 1. [1]
(iii) Find the values of a and b so that f is also differentiable at x = 1, and hence write the complete smooth function. [2]

A civil engineer is modelling the elevation profile of a bridge ramp. The height (in metres) of the ramp at a horizontal distance x (in metres) from the start is described by the function f(x) = (e^(2x) − 1)/sin x for x ≠ 0, and f(0) = k.

A civil engineer is modelling the elevation profile of a bridge ramp. The height (in metres) of the ramp at a horizontal distance x (in metres) from the start is described by the function

f(x) = { (e^(2x) − 1) / (sin x), for x ≠ 0
k, for x = 0 }

(i) For the ramp profile to be physically realizable (no sudden jump in height), the function f must be continuous at x = 0. Find the value of k that ensures this continuity.

(ii) Once k is fixed as found in (i), determine whether f is differentiable at x = 0 by computing f′(0) from first principles (using the definition of the derivative).

A civil engineer models the deflection of a suspension bridge cable using the function y = (sin x)^(cos x) + (cos x)^(sin x), where x ∈ (0, π/2) represents the horizontal position along the bridge deck. To study the rate of change of deflection, the engineer needs to differentiate y with respect to x.

A civil engineer models the deflection of a suspension bridge cable using the function

y = (sin x)^(cos x) + (cos x)^(sin x),

where x ∈ (0, π/2) represents the horizontal position along the bridge deck.

(i) To analyse the rate of change of deflection, the engineer needs dy/dx. Let u = (sin x)^(cos x) and v = (cos x)^(sin x). Find du/dx. [2]

(ii) Using a similar method, find dv/dx, and hence write down dy/dx in simplified form. [2]

A highway engineer is designing a smooth road transition. The elevation profile of the road (in metres) over a stretch is modelled by the function:

f(x) = { ax + b, x ≤ 1
x² + 3, x > 1

where x is the horizontal distance (in km) from a reference point.

A highway engineer is designing a smooth road transition. The elevation profile of the road (in metres) over a stretch is modelled by the function:

f(x) = { ax + b, x ≤ 1
x² + 3, x > 1

where x is the horizontal distance (in km) from a reference point.

(i) For the road profile to have no sudden jumps in elevation, the function f(x) must be continuous at x = 1. Find the condition relating a and b. [1 mark]

(ii) For a smooth gradient transition (no abrupt change in slope), f(x) must also be differentiable at x = 1. Using this condition, find the value of a. [1 mark]

(iii) Using the values found above, determine the value of b, and hence write the complete elevation function f(x). [2 marks]

A civil engineer models the vertical profile of a curved road using the function y = f(x) = √(1 + sin²x), x ∈ ℝ, where y (in metres) represents the height at horizontal position x.

A civil engineer models the vertical profile of a curved road using the function

y = f(x) = √(1 + sin²x), x ∈ ℝ

where y (in metres) represents the height at horizontal position x.

(i) Find f'(x), the rate of change of height with respect to horizontal position. [1]
(ii) Evaluate f'(π/2) and interpret its meaning in context. [1]
(iii) The engineer claims that the road profile satisfies the identity

2y · (dy/dx) = sin 2x

Verify this claim analytically. [2]

If y = log(sin x), then dy/dx is equal to:

The derivative of log(sin x) with respect to x is:

If y = sin x + cos x, find dy/dx.

If f(x) = \begin{cases} \dfrac{\tan 3x}{\sin 2x}, & x \neq 0 \\ k, & x = 0 \end{cases} is continuous at x = 0, then the value of k is:

Differentiate x^(sin x) with respect to x.

If x³ + y³ = 3xy, find dy/dx.

Check whether the function
$$f(x)=\{\begin{array}{ll}\frac{|x-2|}{x-2},& x\ne 2\\ 1,& x=2\end{array}$$
is continuous at $x=2$ or not.

Differentiate sin(x²) with respect to x.

If f(x) = |cos 3x|, find f'(x) at x = π/6.

If x = a(cos t + t sin t) and y = a(sin t − t cos t), find d²y/dx².

Find the value of the constants a and b so that the function f defined below is continuous at x = 1 and x = 3:

f(x) = { 3, if x ≤ 1
ax + b, if 1 < x < 3
7, if x ≥ 3 }

Find the derivative of log(sin²x) with respect to √(cos x).

If x = a(θ − sin θ) and y = a(1 − cos θ), find d²y/dx² at θ = π/2.

A civil engineer models the bending moment of a beam using the function

f(x) = x^(sin x) + (3x² + 2)/(x² + x + 1), x > 0.

The engineer needs to find how rapidly the bending moment changes with position x, and also needs the second component's rate of change separately.

(i) Let u = x^(sin x). By taking logarithms, find du/dx. [2 marks]
(ii) Let v = (3x² + 2)/(x² + x + 1). Find dv/dx and hence find df/dx = d/dx[f(x)]. [2 marks]

A civil engineer is designing a road whose elevation profile is modelled by f(x) = x² sin(1/x) for x ≠ 0, and f(0) = 0. Continuity ensures no sudden jumps in elevation; differentiability ensures no sharp corners in the gradient.

A civil engineer is designing a road that follows a curved path. The elevation profile of the road is modelled by the function:

f(x) = { x² sin(1/x), x ≠ 0
{ 0, x = 0

(i) Check whether f(x) is continuous at x = 0. [1 mark]
(ii) Find f'(x) for x ≠ 0. [1 mark]
(iii) Check whether f(x) is differentiable at x = 0. Hence comment on the smoothness of the road profile at x = 0. [2 marks]

A structural engineer is modelling the deflection of a composite beam using a piecewise function f(x), where x is the horizontal distance (in metres) from one end of the beam. The function must be both continuous and differentiable at the junction point x = 2 to ensure the deflection model is physically realistic and smooth.

A structural engineer is modelling the deflection of a composite beam. The deflection function is given by:

f(x) = { x² + ax + b, if x ≤ 2
{ 3x + 1, if x > 2

where x represents the horizontal distance (in metres) from one end of the beam.

(i) For the deflection to be physically realistic, f must be continuous at x = 2. Find the relationship between a and b. [1 mark]

(ii) The engineer also requires the deflection curve to be smooth (i.e., no sharp bends), which mathematically means f must be differentiable at x = 2. Use this condition to find the value of a. [1 mark]

(iii) Hence find the value of b, and determine the value of f'(1), interpreting what it represents in the context of the problem. [2 marks]

A civil engineer models the elevation profile of a road transition using a piecewise function defined as above. Continuity and differentiability at x = 2 ensure a smooth, safe road transition.

A civil engineer models the elevation profile of a road transition between two sections using the function

f(x) = { kx² + 3x ; 0 < x ≤ 2
{ mx + n ; 2 < x < 5

where x is the horizontal distance (in metres) and f(x) is the elevation (in metres). The road profile must be both continuous and differentiable at the joining point x = 2 to ensure a smooth, safe transition (no sudden change in slope or elevation).

(i) Using the continuity condition at x = 2, write the equation relating k, m, and n. [1]
(ii) Using the differentiability condition at x = 2, find a relationship between k and m. [1]
(iii) If it is further given that the elevation at x = 1 is f(1) = 5 metres, find the values of k, m, and n. [2]

A civil engineer models the vertical displacement y (in metres) of a suspension cable at horizontal distance x (in metres) from a support tower. The relationship is given by:

y = (x + √(x² + 4))³

The engineer needs to analyse the rate of change of this displacement to ensure structural safety standards are met.

A civil engineer is modelling the vertical displacement y (in metres) of a cable at horizontal distance x (in metres) from the support. The displacement is given by:

y = (x + √(x² + 4))³

(i) Find dy/dx. [1 mark]
(ii) Show that dy/dx = 3y · (dy/dx of the inner function) / (x + √(x² + 4)). [1 mark]
(iii) Hence show that (x² + 4)(dy/dx)² = 9y². [2 marks]

A civil engineer is designing a road profile that must be smooth (no sharp bends) to ensure vehicle safety. The height (in metres) of the road above ground level is modelled by the function

f(x) = { x² + 2x + 3, x ≤ 1
{ ax + b, x > 1

where x is the horizontal distance (in km) from the starting point.

A civil engineer is designing a road profile that must be smooth (no sharp bends) to ensure vehicle safety. The height (in metres) of the road above ground level is modelled by the function

f(x) = { x² + 2x + 3, x ≤ 1
{ ax + b, x > 1

where x is the horizontal distance (in km) from the starting point.

(i) Find the value of f(1) and the left-hand limit of f at x = 1. [1]
(ii) For the road to be continuous at x = 1 (no height gap), determine the condition relating a and b. [1]
(iii) For the road to also be smooth (differentiable at x = 1, i.e., no sharp bend), find the values of a and b, and hence write the complete model f(x). [2]

A civil engineer models the slope profile of a pedestrian ramp using a piecewise function f(x). For a smooth, jerk-free transition at the entry point x = 0, continuity of f at x = 0 is essential. The engineer then analyses the logarithm of the slope function to study the rate of change of slope.

A civil engineer is designing a pedestrian ramp for a public building. The slope profile of the ramp is modelled by the function

f(x) = { (√(9 + x) − 3) / x, x ≠ 0
k, x = 0 }

For the ramp to have a smooth, jerk-free transition at the flat entry point x = 0, the slope function must be continuous there.

(i) Find the value of k that makes f continuous at x = 0. [1 mark]
(ii) Once k is fixed, verify that f is continuous at x = 0 by checking the three conditions of continuity. [1 mark]
(iii) The engineer also studies the rate of change of slope. If g(x) = log(f(x)) for x ≠ 0 (using the value of k found in part (i) to extend f smoothly), find the derivative of g(x) = log(√(9 + x) − 3) − log x with respect to x at x = 7. [2 marks]

A civil engineer models the elevation profile of a road using piecewise and exponential functions. Continuity at a join point ensures no abrupt height change, while differentiability of the modified profile determines the smoothness of the road slope.

A civil engineer is designing a road that follows a piecewise-smooth elevation profile. The height (in metres) of the road at a horizontal distance x (in metres) from a reference point is modelled by:

f(x) = { (x² − 4x + 3)/(x − 1), x ≠ 1
{ k, x = 1

For the road to have no abrupt change in elevation (i.e., no sudden jump), the function f(x) must be continuous at x = 1.

(i) Find the value of k so that f(x) is continuous at x = 1. [1 mark]
(ii) Using the value of k found in part (i), state the value of f(1) and verify that lim(x→1) f(x) = f(1). [1 mark]
(iii) After repair work, the engineer modifies the elevation model to:

g(x) = x^(sin x) for x > 0.

Find dg/dx (i.e., g'(x)). [2 marks]

A highway engineer models a road profile using a piecewise function f(x) = ax² + bx + 1 for x ≤ 1, and f(x) = 2x³ − 5 for x > 1. Smoothness of the road requires f to be continuous and differentiable at the join point x = 1.

A highway engineer is designing a smooth transition curve between two road segments. The road profile over a stretch is modelled by the function

f(x) = { ax² + bx + 1, x ≤ 1
{ 2x³ − 5, x > 1

For the road to be 'smooth' (no sudden jerks or breaks), the engineer requires the function f to be both continuous and differentiable at x = 1.

(i) Find the value of 'a' and 'b' so that f(x) is continuous at x = 1. [1 mark]
(ii) Using the values found in (i), verify whether f is differentiable at x = 1. [1 mark]
(iii) The engineer realises that the condition from (i) gives infinitely many pairs (a, b). To make the design unique, she imposes the additional constraint that the slope of the tangent to f at x = 1 equals 6. Find the unique values of a and b, and hence write the complete piecewise function. [2 marks]