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Linear approximation practice problems

Estimate values with the tangent line: L(x) = f(a) + f′(a)(x − a). 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

6 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 3.17

easy
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 49 \) to estimate \( \displaystyle f(\frac{249}{5}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. \sqrt{x} \right|_{\substack{ x=49 }} = 7 \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} \sqrt{x} \right|_{\substack{ x=49 }} = \frac{1}{14} \]
    f'(a).✓ Proved
  4. \[ \frac{x}{14} + \frac{7}{2} \]
    The linearization.Reviewed
  5. \[ \frac{247}{35} \]
    Evaluate at x = \frac{249}{5}.Reviewed
Answer \( L(\frac{249}{5}) = \frac{247}{35} \approx 7.05714 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.18

easy
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{100}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. e^{x} \right|_{\substack{ x=0 }} = 1 \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} e^{x} \right|_{\substack{ x=0 }} = 1 \]
    f'(a).✓ Proved
  4. \[ x + 1 \]
    The linearization.Reviewed
  5. \[ \frac{101}{100} \]
    Evaluate at x = \frac{1}{100}.Reviewed
Answer \( L(\frac{1}{100}) = \frac{101}{100} \approx 1.01 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.20

easy
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(- \frac{1}{20}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. e^{x} \right|_{\substack{ x=0 }} = 1 \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} e^{x} \right|_{\substack{ x=0 }} = 1 \]
    f'(a).✓ Proved
  4. \[ x + 1 \]
    The linearization.Reviewed
  5. \[ \frac{19}{20} \]
    Evaluate at x = - \frac{1}{20}.Reviewed
Answer \( L(- \frac{1}{20}) = \frac{19}{20} \approx 0.95 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.22

easy
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{93}{100}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 0 \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 1 \]
    f'(a).✓ Proved
  4. \[ x - 1 \]
    The linearization.Reviewed
  5. \[ - \frac{7}{100} \]
    Evaluate at x = \frac{93}{100}.Reviewed
Answer \( L(\frac{93}{100}) = - \frac{7}{100} \approx -0.07 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.23

easy
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 8 \) to estimate \( \displaystyle f(\frac{42}{5}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. x^{1 \cdot \frac{1}{3}} \right|_{\substack{ x=8 }} = 2 \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} x^{1 \cdot \frac{1}{3}} \right|_{\substack{ x=8 }} = \frac{1}{12} \]
    f'(a).✓ Proved
  4. \[ \frac{x}{12} + \frac{4}{3} \]
    The linearization.Reviewed
  5. \[ \frac{61}{30} \]
    Evaluate at x = \frac{42}{5}.Reviewed
Answer \( L(\frac{42}{5}) = \frac{61}{30} \approx 2.03333 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.263

easy
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 5 \) to estimate \( \displaystyle f(\frac{503}{100}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. 1 \frac{1}{x} \right|_{\substack{ x=5 }} = \frac{1}{5} \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} 1 \frac{1}{x} \right|_{\substack{ x=5 }} = - \frac{1}{25} \]
    f'(a).✓ Proved
  4. \[ - \frac{x}{25} + \frac{2}{5} = \frac{2}{5} - \frac{x}{25} \]
    The linearization.Reviewed
  5. \[ \frac{497}{2500} \]
    Evaluate at x = \frac{503}{100}.Reviewed
Answer \( L(\frac{503}{100}) = \frac{497}{2500} \approx 0.1988 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More linear approximation practice

Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 2 \) to estimate \( \displaystyle f(\frac{201}{100}) \).
Problem 3.16easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{3}{50}) \).
Problem 3.19easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 2 \) to estimate \( \displaystyle f(\frac{99}{50}) \).
Problem 3.21easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 81 \) to estimate \( \displaystyle f(\frac{402}{5}) \).
Problem 3.24easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(- \frac{7}{100}) \).
Problem 3.25easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(- \frac{3}{50}) \).
Problem 3.26easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{25}) \).
Problem 3.27easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(63) \).
Problem 3.28easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{3}{50}) \).
Problem 3.29easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{27}{25}) \).
Problem 3.30easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{100}) \).
Problem 3.106easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{21}{20}) \).
Problem 3.107easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 9 \) to estimate \( \displaystyle f(\frac{48}{5}) \).
Problem 3.108easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{99}{100}) \).
Problem 3.109easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{2}{25}) \).
Problem 3.110easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{51}{50}) \).
Problem 3.111easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{2}{25}) \).
Problem 3.112easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 10 \) to estimate \( \displaystyle f(\frac{999}{100}) \).
Problem 3.113easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(- \frac{9}{100}) \).
Problem 3.114easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{321}{5}) \).
Problem 3.115easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{19}{20}) \).
Problem 3.116easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(\frac{403}{100}) \).
Problem 3.117easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{3}{100}) \).
Problem 3.118easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{9}{100}) \).
Problem 3.119easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(\frac{401}{100}) \).
Problem 3.120easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 8 \) to estimate \( \displaystyle f(\frac{15}{2}) \).
Problem 3.183easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 9 \) to estimate \( \displaystyle f(\frac{19}{2}) \).
Problem 3.184easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{20}) \).
Problem 3.185easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 125 \) to estimate \( \displaystyle f(\frac{626}{5}) \).
Problem 3.186easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{9}{100}) \).
Problem 3.187easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 16 \) to estimate \( \displaystyle f(\frac{84}{5}) \).
Problem 3.188easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{47}{50}) \).
Problem 3.189easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 8 \) to estimate \( \displaystyle f(\frac{41}{5}) \).
Problem 3.190easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{50}) \).
Problem 3.223easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{53}{50}) \).
Problem 3.224easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 81 \) to estimate \( \displaystyle f(\frac{408}{5}) \).
Problem 3.225easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(5) \).
Problem 3.226easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{7}{100}) \).
Problem 3.227easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 49 \) to estimate \( \displaystyle f(\frac{244}{5}) \).
Problem 3.228easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{50}) \).
Problem 3.229easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(\frac{201}{50}) \).
Problem 3.230easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{641}{10}) \).
Problem 3.264easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{101}{100}) \).
Problem 3.265easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{322}{5}) \).
Problem 3.266easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 27 \) to estimate \( \displaystyle f(\frac{53}{2}) \).
Problem 3.267easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 9 \) to estimate \( \displaystyle f(\frac{47}{5}) \).
Problem 3.268easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 5 \) to estimate \( \displaystyle f(\frac{501}{100}) \).
Problem 3.269easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{637}{10}) \).
Problem 3.270easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{127}{2}) \).
Problem 3.303easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 100 \) to estimate \( \displaystyle f(\frac{498}{5}) \).
Problem 3.304easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 49 \) to estimate \( \displaystyle f(\frac{487}{10}) \).
Problem 3.305easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 16 \) to estimate \( \displaystyle f(\frac{83}{5}) \).
Problem 3.306easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 16 \) to estimate \( \displaystyle f(\frac{82}{5}) \).
Problem 3.307easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{91}{100}) \).
Problem 3.308easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{129}{2}) \).
Problem 3.309easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 9 \) to estimate \( \displaystyle f(8) \).
Problem 3.310easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{97}{100}) \).
Problem 3.343easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 36 \) to estimate \( \displaystyle f(\frac{181}{5}) \).
Problem 3.344easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{107}{100}) \).
Problem 3.345easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = e^{x} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{25}) \).
Problem 3.346easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 10 \) to estimate \( \displaystyle f(\frac{997}{100}) \).
Problem 3.347easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 81 \) to estimate \( \displaystyle f(\frac{403}{5}) \).
Problem 3.348easy✓ Every equation proved
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 100 \) to estimate \( \displaystyle f(\frac{997}{10}) \).
Problem 3.349easy✓ Nihil obstat
Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 8 \) to estimate \( \displaystyle f(\frac{79}{10}) \).
Problem 3.350easy✓ Every equation proved