Problems

In Problems 1-20, determine whether the given vector or scalar is infinitesimal, finite but not infinitesimal, or infinite, (ε, δ are infinitesimal but not zero and H, K are infinite.)

In Problems 21-30, compute the standard part. Assume A, B are real.

In Problems 31-40 determine whether or not the vector has (a) real length, (b) real direction.

41            Prove that st(A + B) = st(A) + st(B).

42            Prove that st(A × B) = st(A) × st(B).

43            Prove that if A is infinite and A - B is finite, then A is almost parallel to B.

44            Prove that if A is finite but not infinitesimal and A - B is infinitesimal, then A is almost parallel to B.

45            Prove that a vector which is parallel to a real vector has a real direction.

The following problems use the notion of a continuous vector valued function. F(f) is said to be continuous at t₀ if each of the components f₁(t), f₂(t), and f₃(t) is continuous at t₀.

46            Prove that F(f) is continuous at f₀ if and only if whenever t ≈ t₀, F(t) ≈ F(t₀).

47            Assume F(t) and G(t) are continuous at t₀. Prove that the following functions are continuous at t₀.

F(t) + G(t), F(t) · G(t), |F(t)|, F(t) × G(t).

48            Prove that if F(t) and h(t) are continuous at t₀, so is h(t)F(t).