Chap 38 Exercises

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Exercise 9 A giant tortoise (with very good eyesight and standing on an unobstructed plane!) spies a head of lettuce on the ground 65 meters away. Being hungry (and knowing the shortest path between two points on the plane!), the tortoise takes off in a straight line for the lettuce. She pretty quickly reaches her top speed, but then starts to tire. If her velocity as a function of time (in meters per minute) is modeled by \(v(t) = 7 t e^{-0.3t}\), how long does it take the tortoise to reach her lunch? Answer this question by finding an calculus/algebra formula for the tortoise’s displacement and then use it to approximate how long it takes to get to the lettuce.

We will be looking at \(\int v(t) dt = 7 \int t e^{-0.3 t} dt\).

We will call the left side of the equation “displacement(t)”. Use integration by parts to find displacement(t) as a simple formula in \(t\).

The tortoise will reach the cabbage at time \(t^\star\) such that \(\text{displacement}(t^\star) = 65\) meters. Use Active R chunk lst-crow-eat-bottle to graph your displacement function to find \(t^\star\). Remember that there will be a constant of integration which can be found by noting that \(\text{displacement}(0) = 0\). (Note that the graphics domain given in the example code isn’t necessarily the best choice for answering the question.)

Active R chunk 2

At what time \(t^\star\) does the tortoise reach the cabbage?

Exercise 10 Which of these candidates for \(f()\) and \(g()\) will produce \[f(g(x))\, \partial_x g(x) = x^3 \cos(x^4)\ ?\]

\(f(x) = \cos(x)/4\) and \(g(x) = x^4\)

\(f(x) = \cos(x)\) and \(g(x) = x^4\)

\(f(x) = x^4\) and \(g(x) = \cos(x)\)

question id: shark-dive-hamper-1

Once again, \(\int h(x) dx = F(g(x))\), where \(\partial_x F(x) = f(x)\).

Exercise 11 Compute the value of the definite integral \[\int_0^1 x \ln(x)\, dx .\] First, compute the anti-derivative using integration by parts, then evaluate the anti-derivative at the lower and upper bounds of integration.

As part of your work, you will need to evaluate \(\left.x^2 \ln(x) {\Large\strut}\right|_{x=0}\). This is an indeterminate form, since \(x=0\) is not in the domain of \(\ln(x)\). So use \(\lim_{x\rightarrow0} \left[{\large\strut} x^2 \ln(x)\right]\) instead Plotting out \(x^2 \ln(x)\) over the domain \(0 < x \leq 10^{-4}\) will guide your answer.

What is the value of \(\int_0^1 x \ln(x)\, dx\)?

Exercise 12 What is \[\int \frac{\sin(x)}{\cos^5(x)}dx\ ?\]

\(\ln(cos(x))\)

\(- \frac{1}{4} \cos^{-4}(x)\)

\(\frac{1}{6} \cos^{-6}(x)\)

question id: No3exI

Exercise 13 Use u-substitution to find \[\int \frac{4 e^{4x} + 4}{e^{4x}}dx\]

Activities

Exercise 15 Using integration by parts, show that \[\int x^2 \, e^{-x}\, dx = e^{-x} \left(2 - 2 x - x^2\right)\] Hint: The 2’s in \(2 - 2x - x^2\) are a hint that you don’t need to integrate \(x^2\).

Show your work.

Exercise 16 The integral \[\int x \sin(x) dx = -x \cos(x) + \sin(x) + C\ .\]

Confirm this result by differentiating both side of the above equation.

As well, use the given result to figure out what was the choice of \(u(x)\) and \(v'(x)dx\) when \(\int x \sin(x) dx\) was integrated by parts.

Show your work.

Exercise 17 Use integration by parts to find \[\int x^2 \ln(x) dx\ .\] Show your choices of \(u(x)\) and \(v'(x) dx\).

Exercise 18 Use integration by parts to find \(\int \ln(x) dx\). Show your work.

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