Determine which of the following functions are linear transformations. For each function, either prove that it is a linear transformation of explain why it is not. (a) S: R² R² given by (b) T: P₂ → P₁ given by I s([*]) = [2+³+1] T(co+c₁x + c₂²) = −c₁ + ₂x. where co, C1, C₂ are constants. (c) F: M2,2 M2,2 given by F(A) = A-AT.

Consider the function f(x) = cos(x2 – 2) + 1 on the interval (-3,3). Calculate the two x-intercepts of this function on the given interval. Express your answers in exact form, as a comma separated list. You may not use the zero/root functionality of a calculator to do this; you must show all of your work and explain your reasoning, where necessary. X =

If f(t)=t and g(t)=sint, determine f ∗g.

In the equation, x and y are functions of t. Differentiate with respect to t to find a relation between dx/dt and dy/dt. x8 + y2 = 1 --------------------------------------------- 8x7 dx dt  + 2y dy dt  = 1   -------------------------------------- 8x7 dx dt  + y2 dy dt  = 1      ------------------------------------------------------------------ 8x8 dx dt  + 2y2 dy dt  = 1   --------------------------------------------------------- 8x8 dx dt  + 2y2 dy dt  = 0   --------------------------------------------------------------------   8x7 dx dt  + 2y dy dt  = 0

Let Pr[A]=0.6, Pr[B]=0.7, and Pr[A u B]=0.8  find Pr[A/B]