Evaluate the following summation: 
${\sum }_{k=1}^4{\left(\frac{k}{{}_{}2}\right)}^2$

A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of

f(x) = √(1 + x⁴)

lies between 0 and 8.

Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1

with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?

You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.

You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?

(Remember that for Simpson’s Rule the number of subintervals must be even.)

Heat capacity of a gas

Heat capacity 

C_v

is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).

The heat capacity of oxygen depends on its temperature T and satisfies the formula

C_v = 8.27 + 10^(-5) * (26T − 1.87T²)

Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for

20°C ≤ T ≤ 675°C.

Evaluate the following summation: 

${\sum }_{i=1}^4{i}^2-3i+8$

Evaluate the following summation: 

${\sum }_{i=0}^2\frac{2i}{3}$

Evaluate the following summation (make sure your calculator is in radian mode): 

${\sum }_{i=1}^4\sin \left(\frac{\pi i}{8}\right)$

Evaluate the following summation: 

${\sum }_{t=2}^{6}5t+\frac{4}{t}$

Evaluate the following summation: 

${\sum }_{i=0}^{4}2i+1$