Local Linearity and Linearization Practice

Sample questions

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1. Sample 1difficulty 2/5

   The linearization of $f(x) = x^3$ at $a = 1$ is

   • A

     $L(x) = 3x$

   • B

     $L(x) = 1$

   • C

     $L(x) = x^3$

   • D

     $L(x) = 1 + 3(x - 1)$

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   Why

   $f(1) = 1$, $f'(1) = 3$. $L(x) = 1 + 3(x-1)$.

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2. Sample 2difficulty 2/5

   The linear approximation of $f$ at $x = a$ is

   • A

     $L(x) = f(x) - f(a)$

   • B

     $L(x) = f(a) + f'(a)(x - a)$

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   • C

     $L(x) = f'(a)x$

   • D

     $L(x) = f(a)$

   Why

   The tangent-line approximation: $L(x) = f(a) + f'(a)(x - a)$.

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3. Sample 3difficulty 2/5

   Use $L(x) = 1 + \tfrac{x}{2}$ (linearization of $\sqrt{1+x}$ at $x = 0$) to estimate $\sqrt{1.04}$.

   • A

     $1.08$

   • B

     $1.02$

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   • C

     $1.20$

   • D

     $1.04$

   Why

   $L(0.04) = 1 + 0.02 = 1.02$.

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4. Sample 4difficulty 2/5

   The differential $dy$ for $y = x^3$ at $x = 2$ is

   • A

     $12\,dx$

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   • B

     $8\,dx$

   • C

     $3\,dx$

   • D

     $6\,dx$

   Why

   $dy = y'\,dx = 3x^2\,dx = 12\,dx$ at $x = 2$.

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5. Sample 5difficulty 2/5

   For $f(x) = e^x$ near $x = 0$, the tangent-line approximation $L(x)$ equals

   • A

     $1$

   • B

     $x$

   • C

     $1 + x$

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   • D

     $e \cdot x$

   Why

   $f(0) = 1$, $f'(0) = 1$. $L(x) = 1 + x$.

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