Begin by preheating your oven to 350 degrees.
In a large bowl add the whole wheat flour, baking soda, white sugar, brown sugar, cinnamon, and nutmeg.
Next add the eggs, vanilla extract, and vegetable oil.
Mix well with a hand-mixer for about three minutes.
Gently mix in the apple pieces.
Add the nuts (if desired).
Pour into a greased 9 x 13 inch pan and bake for 40 to 45 minutes or until a toothpick comes out clean from the center of the cake.