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# Category: Apostol – Calculus 1

Solutions to Calculus Problems

# Find a vector valued function satisfying given properties

If and are fixed vectors and is a vector valued function with find if and .

We have # Prove the zero derivative theorem for vector valued functions

Prove the zero derivative theorem for vector valued functions. In other words, show that if for each in an open interval , then there exists a vector such that for all .

Proof. If then we have By the zero derivative for real valued functions we then have for some constant for each . Therefore we have # Prove a condition for a vector valued function to be differentiable

If is a vector valued function, prove that is differentiable on the open interval if and only if for all we have Proof. First, assume is differentiable on . Then, by the definition of differentiability for vector valued functions we have and each of the exists. Therefore by the usual definition of the derivative we have Conversely, assume that Then, the limit exists for each . Thus, exists for each . Therefore, exists # Prove a property of limits of vector valued functions

Prove that Proof. Let Then, This holds if and only if From the usual limit this is true if and only if This is if and only if # Compute G′ if G = F F′ x F′′

Let be the scalar triple product . Prove that .

Proof. We compute, # Compute G′ if G = F x F′

Let . Compute in terms of and .

Using the product rule for the cross product we compute, Since since a vector crossed with itself is 0.

# Prove that F′′(t) has the same direction as a given F(t)

Let be nonzero vectors and Prove that has the same direction as .

Proof. To show that and have the same direction we must show that there exists a constant such that . So, we compute, # Prove that F”(t) is orthogonal to F'(t) under given conditions

Let be a nonzero vector and a vector valued function with for all , and such that the angle between and is constant. Prove that and are orthogonal.

Proof. Since we have for some constant . Since the angle between and is constant we have for some constant . Therefore, is constant. Hence, is constant, say . So, we have # Compute the dot product of a vector and a vector valued integral

Let and let Compute the dot product .

First, we evaluate the vector valued integral, Therefore, the dot product is # Compute a vector valued integral

Compute the vector valued integral We compute 