F(x) = 1/2 * x ^ 2 + 2x - 5 at x = 2
Evaluate the Derivative at x = 2

4

Step-by-step explanation:

f(x) = 1/2 * + 2x - 5

f'(x) = d/dx (1/2 * + 2x - 5)

=d/dx 1/2 * + d/dx 2x - d/dx 5

d/dx 1/2 * = x  [Because to differentiate a exponential, you multiply the coefficient by the exponent (1/2 * 2) and then subtract 1 from the exponent: 2-1 = 1]

d/dx 2x = 2 [same as the explanation above]

d/dx 5 = 0 [differentiating a constant always equals zero]

so f'(x) = x+2

so f'(2) = 2+2 =4

b: |12 – x| ≤ 0.25

step-by-step explanation:      

let x be the possible lengths in inches, of the ribbons jaime cut.

we have been given that jaime is cutting strips of ribbon for a project. he tries to cut each piece to be 12 inches long, but sometimes he cuts it too long or too short by as much as 0.25 inch.  

we can represent this information in an inequality as:

by the definition of absolute value function we will get,

upon combining our inequalities we will get,

therefore, the inequality   describes all the possible lengths (x), in inches, of the ribbons jaime cut.

step-by-step explanation: