Question

Let C be the curve parametrized by c(t) = (t, t^2/2) for 0 <=t<=1.
1. (a) Calculate the slope of the tangent line to c(t) at t =1/2

Answer 1

$\\ \sf\longmapsto C(t)=(t,\dfrac{t^2}{2})$

$\\ \sf\longmapsto C(1/2)=,\dfrac{1}{2},\dfrac{(1/2)^2}{2})$

$\\ \sf\longmapsto C(1/2)=(\dfrac{1}{2},\dfrac{\dfrac{1}{4}}{2})$

$\\ \sf\longmapsto C(1/2)=(\dfrac{1}{2},\dfrac{1}{8})$

Now

Slope

$\\ \sf\longmapsto \dfrac{y_2-y_1}{x_2-x_1}$

$\\ \sf\longmapsto \dfrac{0-\dfrac{1}{8}}{0-\dfrac{1}{2}}$

$\\ \sf\longmapsto \dfrac{\dfrac{1}{8}}{\dfrac{1}{2}}$

$\\ \sf\longmapsto \dfrac{1}{4}$