Calculus for Engineering Students

A sensor's firmware reports V(x) = (x² − 4)/(x − 2) volts for dimensionless input x ≠ 2, but reports V(2) = 0 volts. An engineer must decide whether measurements near x = 2 indicate an unbounded response or a firmware special-case error.

Which diagnosis is mathematically justified?

• AThe limit is 0 V because the firmware explicitly assigns that output.
• BThe limit does not exist because the formula divides by zero at x = 2.
• CThe limit is 4 V, and changing V(2) to 4 V removes the discontinuity.
• DThe response has a vertical asymptote that assigning V(2) cannot remove.

A valve model specifies Q(p) = 2p L/s for p < 5 bar, Q(5) = 11 L/s, and Q(p) = 2p + 2 L/s for p > 5 bar, with p expressed numerically in bar. The team is assessing behavior at the switching pressure.

Which conclusions are valid? Select all that apply.

• Both one-sided limits exist, but they have different values.
• Changing only Q(5) to 10 L/s would make the model continuous.
• The model has a finite jump rather than an unbounded flow response.
• The model is discontinuous even though Q(5) lies between the one-sided limits.

Four normalized signal models are examined at t = 0: A uses sin(t)/t with A(0) = 1; B equals −1 for t < 0 and 1 for t ≥ 0; C uses 1/t with C(0) = 0; and D uses sin(1/t) with D(0) = 0. The engineer must distinguish bounded switching, unbounded behavior, and persistent oscillation.

Match each signal to its behavior at t = 0.

An electrical component dissipates P(t) = R(t)I(t)². At one instant, I = 2 A, dI/dt = 3 A/s, R = 5 Ω, and dR/dt = 0.5 Ω/s.

What is the instantaneous rate of change of power?

• A60 W/s
• B62 W/s
• C32 W/s
• D2 W/s

A dimensionless actuator gain is G(x) = (1 + x²)³/(2 + x), with x > 0. A sensitivity spreadsheet reports G′/G = 3/(1 + x²) − 1/(2 + x), and its results disagree with small-perturbation simulations.

To correct the first term in G′/G, its numerator 3 must be multiplied by ___.

A hydraulic model gives Q = Δp/R. At one instant, Δp = 100 kPa, R = 20 kPa·s/L, d(Δp)/dt = 10 kPa/s, and dR/dt = 4 kPa/L.

Which calculations are correct at that instant? Select all that apply.

• The flow rate is decreasing at 0.5 L/s².
• The flow rate is increasing at 0.5 L/s².
• Holding R fixed, sensitivity to pressure is 0.05 L/(s·kPa).
• The instantaneous fractional rate Q′/Q is +0.1 s⁻¹.

Water enters an upright conical tank at 0.1 m³/min with no outflow. The tank is 6 m tall with top radius 3 m, and the water depth is currently 2 m.

Assuming a horizontal water surface, how fast is the depth increasing?

• A0.1/(4π) m/min
• B0.1/(2π) m/min
• C0.1/(3π) m/min
• D0.1/π m/min

An aircraft travels horizontally at 100 m/s at a constant altitude of 1,000 m above a radar station. Let x be its signed horizontal displacement from the station and s its distance from the radar.

Assertion: At the instant the aircraft is directly overhead, ds/dt = 0 even though its speed is nonzero. Reason: Constant altitude gives s² = x² + 1000², so ds/dt = (x/s)(dx/dt).

• ABoth A and R are true and R is the correct explanation of A
• BBoth A and R are true but R is NOT the correct explanation of A
• CA is true but R is false
• DA is false but R is true

Four systems change smoothly: a fixed-area piston of 0.02 m² receives 0.004 m³/s; a sphere has r = 2 m and r′ = 0.1 m/s; a straight cable has perpendicular endpoint separations x = 3 m and y = 4 m with x′ = 2 m/s and y′ = 0; and a rectangle has L = 2 m, W = 3 m, L′ = 0.1 m/s, and W′ = −0.2 m/s. There are no leaks or geometric deformations beyond those stated.

Match each requested geometric rate to its value.

An open-top tank has a square base of side x meters and must hold 1 m³. With equal material cost per unit area, its material objective is S(x) = x² + 4/x square meters for x > 0.

Which design conclusions are justified? Select all that apply.

• Without an upper bound on x, the minimum occurs at x = ∛2 m.
• Because S″(x) > 0 for every x > 0, its stationary point is a global minimum.
• The minimum-material tank must have equal base side and height.
• If manufacturing additionally requires x ≤ 1 m, the minimum occurs at x = 1 m.

A fan's hourly operating penalty is C(v) = v³ − 3v² − 9v + 30, where v is a dimensionless speed setting constrained to 0 ≤ v ≤ 2. An unconstrained calculation finds a stationary point at v = 3 with C″(3) > 0.

Which setting globally minimizes the penalty over the allowed range?

• Av = 2, where the feasible penalty is 8.
• Bv = 3, where the positive second derivative certifies the minimum.
• Cv = 0, because the lower speed bound minimizes operating effort.
• Dv = 1, where the zero second derivative identifies the minimum.

A simplified heat-sink design uses positive normalized dimensions a and b with ab = 4 and 1 ≤ a ≤ 4. Its thermal penalty is T = a + b, and the team must certify a global rather than merely stationary optimum.

Arrange the calculation steps in a valid dependency order.

A test carriage has velocity v(t) = 3t² − 6t m/s for 0 ≤ t ≤ 3 s, with t entered in seconds. The controller report must distinguish net displacement from total distance traveled.

What are the displacement and total distance, respectively?

• A0 m and 0 m
• B8 m and 8 m
• C0 m and 4 m
• D0 m and 8 m

A slowly driven actuator moves from x = 0 to x = 2 m against a resisting force F(x) = 100x + 50x² N, with x entered in meters. The team is comparing force-based work calculations.

Which calculations correctly describe the required work? Select all that apply.

• Integrating F over displacement gives W = 1000/3 J.
• The displacement-averaged force is 500/3 N, giving the same work when multiplied by 2 m.
• Using the final force throughout the stroke gives the exact work, W = 800 J.
• Averaging only the two endpoint forces gives the exact work, W = 400 J.

A vertical gate is 2 m wide and extends from 1 m to 3 m below the water surface in an open tank. Its dry face and the water surface are exposed to the same atmospheric pressure, but a spreadsheet integrates atmospheric pressure plus ρgy over the wet face and labels that result the net gate load.

To obtain the net load without counting the atmospheric contribution that cancels between the two faces, the pressure-area integral should use ___.

A flow dataset gives Q = 1, 3, and 9 m³/s at t = 0, 1, and 2 s. A trusted reference gives accumulated volume 22/3 m³ over this interval, and the acceptance requirement is less than 1% relative error.

Which statements about implementations on these samples are correct? Select all that apply.

• Composite trapezoidal integration gives 8 m³.
• Both trapezoidal and Simpson integration meet the 1% requirement.
• Simpson’s 1/3 rule gives 22/3 m³ and meets the requirement.
• The trapezoidal relative error is approximately 9.09%.

A logger records Q = 0, 2, and 6 L/s at t = 0, 1, and 3 s. A calibration run establishes that Q(t) = 2t L/s throughout the interval, so the reference volume is 9 L.

Which spreadsheet calculation is valid and meets the error requirement?

• AUse Simpson’s standard weights with average spacing 1.5 s to obtain 7 L.
• BUse Simpson’s standard weights with spacing 1 s to obtain 14/3 L.
• CUse trapezoids with widths 1 s and 2 s to obtain 9 L.
• DMultiply the arithmetic mean of all three samples by 3 s to obtain 8 L.

Four integration scripts are audited: A uses trapezoids on t = [0,1,2], y = [0,1,4], with reference 8/3; B uses Simpson’s 1/3 rule on the same data; C applies composite Simpson’s 1/3 rule alone across all three intervals of t = [0,1,2,3], y = [0,1,8,27]; D uses one average spacing in the equal-spacing trapezoidal formula for t = [0,1,3], y = [0,1,9]. Each script must use a valid formula and achieve less than 1% relative error.

Match each script to its audit finding.

A proposed response model uses the infinite series Σₙ₌₀∞ n!xⁿ. At x = 0.1, an engineer observes that several early terms decrease and uses this as evidence for convergence.

Assertion: The series converges at x = 0.1 because its first several terms decrease. Reason: For a series to converge, its individual terms must approach zero.

• ABoth A and R are true and R is the correct explanation of A
• BBoth A and R are true but R is NOT the correct explanation of A
• CA is true but R is false
• DA is false but R is true

Firmware uses a Taylor polynomial P(x) for gain f(x) = e^(−x), and a verified remainder bound guarantees |f(x) − P(x)| ≤ 10⁻⁴ over the operating range. The product requirement is 1% error relative to the true gain, including operation at x = 10.

The certificate directly bounds ___, so it does not by itself establish the required percentage accuracy at x = 10.

A controller approximates ln(1 + x) at x = 0.5 using its Maclaurin series x − x²/2 + x³/3 − ⋯. The absolute-error requirement is less than 0.01.

Which conclusions are justified? Select all that apply.

• The three-term polynomial meets the requirement because its next-term bound is below 0.01.
• The four-term polynomial meets the requirement because its next-term bound is 0.00625.
• Integrating the convergent geometric series for 1/(1 + x) gives a valid construction of this series at x = 0.5.
• Convergence at x = 0.5 guarantees that every nonzero truncation meets the requirement.

A well-mixed tank initially contains 100 L of pure water. Brine enters at 2 L/min with salt concentration 3 g/L, and mixture leaves at 2 L/min, keeping the volume constant.

Which expression gives the salt mass M(t) in grams, with t in minutes?

• AM(t) = 300(1 − e^(−0.02t))
• BM(t) = 6t
• CM(t) = 300e^(−0.02t)
• DM(t) = 6(1 − e^(−0.02t))

A cooling component is modeled by T′ = −0.1(T − 20), with time in minutes and temperatures in degrees Celsius. Its initial temperature is 80 °C.

Assertion: T(t) = 20 + 60e^(−0.1t) satisfies both the differential equation and the initial condition. Reason: The exponential solution is valid only if both the component and ambient temperatures are converted to kelvin.

• ABoth A and R are true and R is the correct explanation of A
• BBoth A and R are true but R is NOT the correct explanation of A
• CA is true but R is false
• DA is false but R is true

Four first-order models use time t in seconds: capacitor voltage V′ = 2(5 − V), V(0) = 0; tank solute mass M′ = 4 − 0.5M, M(0) = 10; drag-limited speed v′ = 6 − 3v, v(0) = 0; and tracer amount N′ = −0.2N, N(0) = 100. All coefficients carry the units required by their respective equations.

Match each initial-value model to the solution that satisfies both its rate equation and initial condition.

A gas model is P = CT/V with C = 20 kPa·m³/K, evaluated at T = 300 K and V = 2 m³. The team considers a specified change ΔT = +1 K, ΔV = +0.01 m³ and, separately, independent measurement standard uncertainties σT = 1 K and σV = 0.01 m³.

Using first-order sensitivity analysis, which statements are correct? Select all that apply.

• The specified simultaneous changes give an estimated pressure change of −5 kPa.
• The independent standard uncertainties give σP = √325 kPa.
• The independent standard uncertainties cancel to give σP = 5 kPa.
• If the same magnitudes are treated as hard error bounds, the worst-case linear pressure-error bound is 25 kPa.

A plate has T(x,y) = 20 + x² + xy + 2y² °C when x and y are entered in meters. At (1,1) m, a probe moves in the unit direction u = (3/5,4/5).

What is the instantaneous temperature change per meter traveled in that direction?

• A8 °C/m
• B29/5 °C/m
• C√34 °C/m
• D29 °C/m

Resistance is modeled as R = ρL/A with known constant resistivity. A shared calibration effect makes measured L and A positively correlated, but an engineer propagates uncertainty by adding only the squared individual sensitivity contributions.

The first-order variance formula must also include their ___, whose contribution can reduce the resistance variance because the two sensitivities have opposite signs.

A solid cylinder has radius 2 m, height 3 m, and density ρ(r) = kr, where r is distance from its axis and k = 1 kg/m⁴. An analyst proposes cylindrical coordinates for mass and axial moment of inertia.

Which statements are correct? Select all that apply.

• The mass is 12π kg because the density's factor r already supplies the coordinate weighting.
• The mass is 16π kg after including both density and the cylindrical volume element.
• The axial moment of inertia is 32π kg·m² by applying I = MR²/2.
• The density's factor r does not replace the separate factor r in dV.

An engineer needs the mass of a uniform disk of radius a and surface density σ, the mass of a uniform solid sphere of radius a and volume density ρ₀, the mass of a uniform rectangular block with sides a, b, c, and the axial moment of inertia of a uniform cylindrical sleeve a ≤ r ≤ 2a with height H. All lengths and densities are positive.

Match each requested quantity to an appropriate coordinate integral.

A triangular lamina has vertices (0,0), (2,0), and (0,1) m and surface density σ(x,y) = 1 + x kg/m² when x is entered in meters. The team needs the x-coordinate of its center of mass, accounting for the nonuniform density.

Arrange the calculation steps in dependency order.

A force field is F(x,y) = (−y,x) N when coordinates are entered in meters. A particle follows the radius-2 m quarter circle from (2,0) to (0,2), counterclockwise.

What work does the field perform along this oriented path?

• A0 J
• B4π J
• C−2π J
• D2π J

A uniform fluid velocity is v = (0,0,2) m/s through a planar panel of actual area 3 m². Its chosen unit normal is n = (0,3/5,4/5).

Which flux statements are correct? Select all that apply.

• The signed volume flux is 24/5 m³/s.
• The flux is 6 m³/s because speed times actual area always gives flux.
• Reversing the chosen normal gives flux −24/5 m³/s.
• Using the panel's horizontal projected area, 12/5 m², gives the same positive flux.

Two simulators integrate F(x,y) = (y,x) N, with coordinates entered in meters, from (0,0) to (1,1) along different smooth paths in the plane. They report 1 J and 0.8 J, and a reviewer attributes the difference to genuine path dependence.

The discrepancy instead indicates numerical or implementation error because this force field is ___ on the entire plane.

A normalized radial field is F(x,y,z) = (x,y,z)/(x² + y² + z²)^(3/2), undefined at the origin. An engineer wants its outward flux through the unit sphere and notes that ∇·F = 0 away from the origin.

Assertion: The outward flux through the unit sphere is 4π. Reason: The divergence of F vanishes at every nonzero point.

• ABoth A and R are true and R is the correct explanation of A
• BBoth A and R are true but R is NOT the correct explanation of A
• CA is true but R is false
• DA is false but R is true

For the normalized planar field F = (−y/(x² + y²), x/(x² + y²)), an engineer considers the annulus 1 ≤ x² + y² ≤ 4. Its positively oriented boundary consists of the outer circle counterclockwise and the inner circle clockwise.

Which uses of Green’s theorem and orientation are valid? Select all that apply.

• Green’s theorem applies to the annulus because the field is smooth on a neighborhood of that region.
• The outer circle alone has zero circulation because the curl vanishes throughout the annulus.
• The sum of the circulations around the two stated boundary components is zero.
• Reversing only the inner circle leaves the boundary circulation unchanged because the curl is zero.

In normalized coordinates, F = (−y/2,x/2,z) is smooth everywhere. The chosen surface is the upper hemisphere x² + y² + z² = a², z ≥ 0, oriented outward, and the team wants circulation around its induced boundary.

Which applications of Stokes’ theorem are correct? Select all that apply.

• The boundary circulation is πa².
• The induced boundary direction is counterclockwise when viewed from positive z.
• The hemisphere may be replaced by the upward-oriented flat disk with the same boundary.
• The circulation is 2πa² because the curl has magnitude one and the hemisphere has area 2πa².