Chapter 6.1

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Take \(f\) to be the function that is given by \[f(x)=5x+2+E_{-1}(x)\] where \(E_{-1}\) is \(o(x+1)\). Determine \(f'(-1)\) and determine an equation for the line \(L\) that is tangent to \(f\) at \((-1,f(-1)).\)
  2. Take \(f(x)=x^3.\) Write the local linear approximation of \(f\) at \(a=1.\) Use your local linear approximation to approximate \((1.05)^3.\)
  3. The following functions are continuous at \(x_0=1\), but are not differentiable at \(x_0=1\). Explain why.
    1. \(f(x)=|x-1|\)
    2. \(f(x)=\sqrt{x-1}\)
  4. Calculate the derivative of each function by decomposing it into a sum and or product of simpler functions and by using the appropriate derivative rule.
    1. \(f(x)=5x^4+\sin(x)-\mathrm{e}\)
    2. \(f(x)=x^{4/5}\exp_5(x)+\sqrt{x}\tan(x)\)
  5. Calculate the derivative of each function \(f.\)
    1. \(f(x)=\tfrac{1}{\sin(x)}\)
    2. \(f(x)=\frac{x+3}{\ln(x)}\)
    3. \(f(x)=\exp(x)\cdot \frac{2x}{\exp(x)+\cos(x)}\)
  6. Calculate the derivative of each function \(f.\)
    1. \(f(x)=\sqrt{4\cos(x)+\ln(x)}\)
    2. \(f(x)=\sin(\mathrm{e}^x)\cos(3x+1)\)
  7. For the function \(f\) given below, decompose \(f\) into simpler functions in order to find a formula for \(f'(x):\) \[f(x)=(3x+5)^4\sqrt{x^5-x}+g(3x+1)+(g(x))^3,\] where \(g\) is a differentiable function. Your answer may involve \(g'\) and \(g.\)
  8. Use Newton’s Method to approximate the value \(5^\frac{1}{5}.\) Start with an initial guess of \(x_1=1\) and apply the method three times.

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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