Quiz 1: Derivatives and Taylor Series

A concrete beam under loading exhibits a deflection curve described by 𝑦(𝑥). The first derivative 𝑑𝑦/𝑑𝑥 at a point along the beam represents:

• The maximum bending stress at that section

• The curvature of the beam at that section

• The slope or angle of deflection at that point

• The total strain energy in the beam

In highway design, the first derivative of elevation along a road profile indicates:

• The change in curvature per unit distance

• The slope or grade of the road

• The total elevation gain

• The radius of the curve

The second derivative of the deflection curve 𝑦(𝑥) in a simply supported beam represents:

• Shear force

• Bending moment

• Curvature

• Axial stress

A surveyor uses Taylor Series to approximate a terrain elevation function near a known point. What is the physical significance of truncating the series after the first derivative term?

• Assuming the slope remains constant

• Ignoring all variations in curvature

• Both A and B

• None of the above

If the velocity of a falling object is represented by the first derivative of its displacement, what does the second derivative signify in a dam spillway study?

• Velocity of water flow

• Rate of energy loss

• Acceleration due to gravity

• Flow discharge

In pavement design, a Taylor Series expansion is used to estimate the stress at a point beneath the surface. Higher-order terms in the expansion help improve accuracy because they:

• Account for linear variations only

• Include nonlinear behavior of stress distribution

• Reduce computational time

• Ignore soil nonlinearity

A structural engineer uses the Taylor Series to approximate deflection 𝑦(𝑥) near the midspan of a beam. Neglecting higher-order terms causes:

• Underestimation of bending moment

• Overestimation of curvature

• Small truncation error depending on the order of approximation

• No effect at all

The derivative 𝑑𝑣/𝑑𝑥 of a velocity distribution in a river cross-section indicates:

• Average velocity

• Flow depth

• Shear rate or velocity gradient

• Energy head

In finite difference approximation, replacing a derivative by a difference quotient is equivalent to:

• Expanding a function by Taylor Series and neglecting higher terms

• Using exact differentiation rules

• Linear interpolation of data

• Applying boundary conditions only

If the displacement of a pile under load 𝑦(𝑡) is measured over time, the first derivative 𝑑𝑦/𝑑𝑡 physically means:

• Rate of settlement

• Total settlement

• Elastic modulus

• Factor of safety