Question

Help calculus module 6 DBQ

please show work

Answer 1

1. Let
`a,b,c` be the three points of intersection, i.e. the solutions to
`f(x)=g(x)`. They are approximately

`a\approx-3.638`

`b\approx-1.862`

`c\approx0.889`

Then the area
`R+S` is

`\displaystyle\int_a^c|f(x)-g(x)|\,\mathrm dx=\int_a^b(g(x)-f(x))\,\mathrm dx+\int_b^c(f(x)-g(x))\,\mathrm dx`

since over the interval
`[a,b]` we have
`g(x)\ge f(x)`, and over the interval
`[b,c]` we have
`g(x)\le f(x)`.

`\displaystyle\int_a^b\left(\frac{x+1}3-\cos x\right)\,\mathrm dx+\int_b^c\left(\cos x-\frac{x+1}3\right)\,\mathrm dx\approx\boxed{1.662}`

2. Using the washer method, we generate washers with inner radius
`r_(\rm in)(x)=2-\max\{f(x),g(x)\}` and outer radius
`r_(\rm out)(x)=2-\min\{f(x),g(x)\}`. Each washer has volume
`\pi({r_(\rm out)(x)}^2-{r_(\rm in)(x)}^2)`, so that the volume is given by the integral

`\displaystyle\pi\int_a^b\left((2-\cos x)^2-\left(2-\frac{x+1}3\right)^2\right)\,\mathrm dx+\pi\int_b^c\left(\left(2-\frac{x+1}3\right)^2-(2-\cos x)^2\right)\,\mathrm dx\approx\boxed{18.900}`

3. Each semicircular cross section has diameter
`g(x)-f(x)`. The area of a semicircle with diameter
`d` is
`\frac{\pi d^2}8`, so the volume is

`\displaystyle\frac\pi8\int_a^b\left(\frac{x+1}3-\cos x\right)^2\,\mathrm dx\approx\boxed{0.043}`

4.
`f(x)=\cos x` is continuous and differentiable everywhere, so the the mean value theorem applies. We have

`f'(x)=-\sin x`

and by the MVT there is at least one
`c\in(0,\pi)` such that

`-\sin c=(\cos\pi-\cos0)/(\pi-0)`

`\implies\sin c=\frac2\pi`

`\implies c=\sin^(-1)\frac2\pi+2n\pi`

for integers
`n`, but only one solution falls in the interval
`[0,\pi]` when
`n=0`, giving
`c=\sin^(-1)\frac2\pi\approx\boxed{0.690}`

5. Take the derivative of the velocity function:

`v'(t)=2t-9`

We have
`v'(t)=0` when
`t=\frac92=4.5`. For
`0\le t<4.5`, we see that
`v'(t)<0`, while for
`4.5<t\le8`, we see that
`v'(t)>0`. So the particle is speeding up on the interval
`\boxed{\frac92<t\le8}` and slowing down on the interval
`\boxed{0\le t<\frac92}`.