Chapter 6.3

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Use Fermat’s theorem to determine all points at which each function \(f\) below can potentially attain a local maximum or a local minimum.
    1. \(f(x) = -4 x^2 + 4 x + 3\)
    2. \(f(x)=x^{2/3}+x\)
  2. Take \(f\) to be the function that is given by \(f(x) = -4 x^2 + 4 x + 3\). Determine the maximum and minimum values that \(f\) attains on \([0,2].\)
  3. Take \(f(x)=\tfrac{1}{4}x^2+\frac{1}{2}.\) This is a differentiable function on the interval \((-2, 1)\) and a continuous function on the interval \([-2,1].\)
    1. Find a value \(M\) so that \(|f'(x)|\leq M\) for all \(x\) in \([-2,1].\)
    2. Given that \(f(-2)=\tfrac{3}{2}\), find the smallest range of values that is guaranteed to contain \(f([-2, 1])\) and sketch the smallest region that you can that is guaranteed to contain \(f\). Then compare it to the graph of \(f\).
    3. Given that \(f(-2)=\tfrac{3}{2}\) and \(f(1)=\tfrac{3}{4}\), find the smallest range of values that is guaranteed to contain \(f([-2, 1])\) and sketch the smallest region that you can that is guaranteed to contain \(f\). Then compare it to the graph of \(f\).
  4. Take \(f\) and \(g\) to be differentiable functions on the interval \([1,9]\) so that \(f(2)=4\), \(g(2)=8\), and that for any \(x\) in \((1,9),\) \[f'(x)-g'(x)=0.\] Determine \(g(9)-f(9).\)
  5. For each function \(f\) that is given below, determine the antiderivative of \(f:\)
    1. \(f(x) = 20x^3 + e^x +2^{x}+ 4\cos(x)\)
    2. \(f(x) = \frac{3}{x}+\frac{40}{x^2+1} - \ln(2)\sqrt{x}+2\)
  6. Take \(f\) to be the function with the property that \[\int f(x)\,\mathrm{d}x=\sin(x)+\tan(x)+2x^3+C.\] Identify the function \(f\).
  7. Calculate the following:
    1. \(\displaystyle\int_{1}^2x^5\,\mathrm{d}x\)
    2. \(\displaystyle\int_{0}^1 ( 20x^3 + e^x +2^{x}+ 4\cos(x))\,\mathrm{d}x\)
  8. Determine the following.
    1. \(\displaystyle \int 4x^3\sin(x^4)\,{\rm d}x\)
    2. \(\displaystyle \int (40x^3+x)\sqrt{20x^4+x^2+1}\,{\rm d}x\)
    3. \(\displaystyle \int \frac{e^x+\ln(2)2^x}{e^x +2^x}\,{\rm d}x\)
  9. Use the fact that \[(x\cos(x))'=\cos(x)-x\sin(x)\] to determine \[\int x\sin(x)\,\mathrm{d}x.\]
  10. Verify that the following are of indeterminate forms and use L’Hospital’s rule to determine the given limits:
    1. \(\displaystyle \lim_{x\to 5}\frac{x^3 - 24 x - 5}{x-5}\)
    2. \(\displaystyle \lim_{x\to \infty}\left(\sqrt{4x^2+3x +1} -2x\right)\)
    3. \(\displaystyle \lim_{x\to \infty} \frac{x^3}{{\rm e}^x}\)

Practice Problems

Return

© Copyright 2024 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart.

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.