Further Insights and Challenges

Question 13.125

Let r(t) = 〈x(t), y(t)〉 trace a plane curve . Assume that x′(t0) ≠ 0. Show that the slope of the tangent vector r′(t0) is equal to the slope dy/dx of the curve at r(t0).

Question 13.126

Prove that .

Question 13.127

Verify the Sum and Product Rules for derivatives of vector-valued functions.

Question 13.128

Verify the Chain Rule for vector-valued functions.

Question 13.129

Verify the Product Rule for cross products [Eq. (5)].

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Question 13.130

Verify the linearity properties

Question 13.131

Prove the Substitution Rule (where g(t) is a differentiable scalar function):

Question 13.132

Prove that if ∥r(t)∥ ≤ K for t ∈ [a, b], then