Lecture Nineteen Practice

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Practice problems for Lecture Nineteen Content

1 : Compute the differential of the function $y=-\frac {5}{{\left (x + 3\right )}^{2}}$ .

$dy = \answer {\left (\frac {10}{{\left (x + 3\right )}^{3}}\right )\,dx}$

2 : Compute $dy$ using the function $y=-\frac {4}{x - 2}$ as $x$ goes from $3$ to $2.9$ .

$dy =\answer {-0.4}$

3 : Compute $dy$ using the function $y=4 \, x^{2}$ as $x$ goes from $-1$ to $-1.1$ . $dy = \answer {0.8}$

4 : Compute $\Delta y$ using the function $y=5 \, x^{3}$ as $x$ goes from -2 to 1. $\Delta y = \answer {45}$

5 : Compute the differential of the function $y=\cos \left (x + 1\right )$ . $dy = \answer {\left (-\sin \left (x + 1\right )\right )\,dx}$

6 : Compute $dy$ and $\Delta y$ of the function $y=\sqrt {x - 4}$ as $x$ goes from $5$ to $5.2$ . $dy = \answer {0.1}$ $\Delta y = \answer {\sqrt {1.2}-1}$

7 : Find the linear approximation of the function $f(x) = \sqrt {x + 3}$ at $a = 22$ and use it to approximate the numbers $(24.9)^{\frac {1}{2}}$ and $(25.01)^{\frac {1}{2}}$ . $L(x) = \answer {\frac {1}{10} \, x + \frac {14}{5}}$

7.1 : Round to 3 decimal places $\left (24.9\right )^{\frac {1}{2}} \approx \answer {4.99}$

7.1.1 : Round to 3 decimal places $\left (25.01\right )^{\frac {1}{2}} \approx \answer {5.001}$

8 : Find the linear approximation of the function $f(x) = \sqrt {x - 3}$ at $a = 4$ and use it to approximate the numbers $(0.900)^{\frac {1}{2}}$ and $(1.010)^{\frac {1}{2}}$ . $L(x) = \answer {\frac {1}{2} \, x - 1}$

8.1 : Round to 3 decimal places $\left (0.900\right )^{\frac {1}{2}} \approx \answer {0.95}$

8.1.1 : Round to 3 decimal places $\left (1.010\right )^{\frac {1}{2}} \approx \answer {1.005}$

9 : Approximate $e^{0.2}$ by letting $f(x) = e^x$ and $a = 0$ . $e^{0.2} \approx \answer {1.2}$