Quiz Chapter 02.02: Differentiation of Continuous Functions

1. The definition of the first derivative of a function f \left( x \right) is

2. The exact derivative of f \left( x \right) = x^{3} at x = 5 most nearly is

3. Using the forward divided difference approximation with a step size of 0.2, the derivative of f \left( x \right) = 5e^{2.3x} at x=1.25 most nearly is

4. A student finds the numerical value of \dfrac{d}{dx} \left( e^{x} \right) = 20.220 at x=3 using a step size of 0.2. Which of the following methods did the student use to conduct the differentiation?

5. Using the backward divided difference approximation, \dfrac{d}{dx} \left( e^{x} \right)=4.3715 at x=1.5 for a step size of 0.05. If you keep halving the step size to find \dfrac{d}{dx} \left( e^{x} \right) at x=1.5 before two significant digits can be considered to be at least correct in your answer, the step size would be (you cannot use the exact value to determine the answer)

6. The heat transfer rate q over a surface given by q=-kA \dfrac{dT}{dy} where

• • • k= thermal conductivity \left( \frac{J}{s-m-K}\right)
    • A= surface area \left( m^{2} \right)
    • T= temperature \left( K \right)
    • y= distance normal to the surface

and the temperature T over the surface varies as

• • • T=-1493 y^{3} + 2200 y^{2} - 1076 y + 500
    • k=0.025 \, \frac{J}{s-m-K}
    • A=3 \, m^{2}

The heat transfer rate q in Watts at the surface most nearly is