Use the graph of f(t) = 2t + 5 on the interval [-1, 4] to write the function F(x), where f of x equals the integral from 1 to x of f of t dt

Question

Use the graph of f(t) = 2t + 5 on the interval [-1, 4] to write the function F(x), where  

A)F(x) = 2x + 5

B)F(x) = x² + 5x - 6

C)F(x) = x² + 5x - 1

D)F(x) = x² + 6x

Answer 1

Using the graph of f(t) = 2t+5, we want the area under f(t) for the interval [1,x]. This is the area of a large triangle with base of x - (-2.5) and height of 2x+5 subtracting the area of the smaller triangle with base 1 - (-2.5) and height of f(1)=7. Area of a triangle is base×height÷2.

F(x) = (x+2.5)(2x+5)/2 - (3.5)(7)/2

= (2x²+10x + 12.5)/2 - 24.5/2

= x²+5x+6.25 - 12.25

= x²+5x-6 (Answer B)

This is identical to

F(x) = ∫₁^x 2t+5 dt = t²+5t|₁^x = x²+5x - (1+5) = x²+5x-6 (Answer B)

Answer 2

Check it out!