### Home > PC3 > Chapter 13 > Lesson 13.3.5 > Problem13-152

13-152.

Sketch the velocity function given by $v\left(t\right) = 6t$ for $0\le t\le5$.

1. Write an expression for the position from $t = 0$ to any value of $t$.

The position function is the area under the curve of the velocity function.
In this case, the area under the velocity curve is a triangle.

area under $v\left(t\right): 0.5\left(\text{base}\right)\left(\text{height}\right) = 0.5\left(t\right)\left(6t\right)$

2. Given the position expression from part (a), write an expression for the slope of the tangent line at any point $t$.

$\lim \limits_{h \to 0}\frac{3(t+h)^{2}-3{t}^2}{h}$

3. How is the expression in part (b) related to $v\left(t\right)$?