The trapezoid rule states that 

$\displaystyle \\ A\approx \frac{\Delta x}{2}[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)]\\ \Delta x=\frac{b-a}{n}, x_i=a+i\Delta x$.

Therefore, using our graph, we have:

$\displaystyle \\ \Delta x=\frac{8}{4}=2\\ \\ x_i=0,2,4,6,8$

We find the function values at the sample point:

$\displaystyle f(x_i)=6,4,4,2.4,2.8$

Then we substitute the appropriate values into the trapezoid rule approximation:

$\displaystyle A\approx\frac{2}{2}[6+2(4+4+2.4)+2.8]=29.6$

Is the following function increasing or decreasing at the point $\displaystyle x=-6$?

$\displaystyle h(x)=-x^5+5x^4+2x$

Increasing and decreasing intervals can be found via the first derivative. Since derivatives measure rates of change, the sign of the derivative at a given point can tell you whether a function is increasing or decreasing.

Begin by taking the derivative of our function:

$\displaystyle h(x)=-x^5+5x^4+2x$

Becomes:

$\displaystyle h'(x)=-5x^4+20x^3+2$

Next, find h'(-6) and look at the sign.

$\displaystyle h'(x)=-5(-6)^4+20(-6)^3+2=-43198$

So, our first derivative is very negative at the given point. This means that h(x) is decreasing.

The trapezoidal approximation of a definite integral is given by the following formula:

$\displaystyle \int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get

$\displaystyle 5(\frac{36}{2})=90$

To evaluate a definite integral using the trapezoidal approximation, we must use the formula

$\displaystyle \int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get

$\displaystyle 2(\frac{27+9+10-1-1-6}{2})=46$

To evaluate a definite integral using the trapezoidal approximation, we must use the formula

$\displaystyle \int_{a}^{b}f(x)dx= (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get

$\displaystyle \pi(\frac{-3-3}{2})=-3\pi$

To evaluate the integral using the trapezoidal rule, we must use the formula

$\displaystyle \int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get the following:

$\displaystyle 2\pi(\frac{e^{4\pi}-1}{10})$

To evaluate a definite integral using the trapezoidal approximation, we must use the following formula:

$\displaystyle \int_{a}^{b} f(x)dx\approx \frac{(b-a)(f(a)+f(b))}{2}$

So, using the above formula, we get

$\displaystyle \frac{(5)(100+e^5+26)}{2}$

which simplifies to

$\displaystyle \frac{5}{2}(126+e^5)$

To evaluate the definite integral using the trapezoidal approximation, we must use the following formula:

$\displaystyle \int_{a}^{b}f(x)dx\approx (b-a)\frac{(f(a)+f(b))}{2}$

Using the above formula, we get

$\displaystyle \frac{2(\frac{48}{4}+\frac{4}{2})}{2}=14$

To evaluate the integral using the trapezoidal approximation, we must use the following formula:

$\displaystyle \int_{a}^{b} f(x)dx\approx (b-a)\frac{(f(b)+f(a))}{2}$

Using the formula, we get

$\displaystyle \frac{4(\frac{5}{6}+\frac{4+e^{-4}}{-6})}{2}=\frac{1-e^{-4}}{3}$

To evaluate the definite integral using the trapezoidal approximation, the following formula is used:

$\displaystyle \int_{a}^{b}f(x)dx\approx (b-a)\frac{(f(b)+f(a))}{2}$

Using the above formula, we get

$\displaystyle \frac{10[(\frac{100}{5}+50)+(0)]}{2}=350$.