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Exercises and solutions from textbooks

# Compute the vector valued integral

Compute the vector valued integral

We compute,

# Compute the integral of (sin t, cos t, tan t)

Compute the vector-valued integral

We compute

# Compute the integral of (t, t1/2, et)

Compute the vector-valued integral

We compute,

# Prove that the angle between F(t) and F'(t) is constant for a given vector function

Consider the vector-valued function

Prove that the angle between and is a constant.

Proof. First, we have

Thus,

Hence, and are orthogonal, so the angle between them is constant,

# Compute derivatives of F(t) = log (1+t2) i + arctan t j + 1/(1+t2) k

Compute the derivatives and of the vector valued function

We compute

# Compute derivatives of F(t) = cosh t i + sinh (2t) j + e-3tk

Compute the derivatives and of the vector valued function

We compute

# Compute derivatives of F(t) = 2eti + 3etj

Compute the derivatives and of the vector valued function

We compute,

# Compute derivatives of F(t) = (arcsin t, arccos t)

Compute the derivatives and where is the vector valued function

We compute

# Compute derivatives of F(t) = (cos t, sin2 t, sin (2t), tan t)

Compute the derivatives and of the vector valued function

We compute

# Compute the first two derivatives of F(t) = (t, t2, t3, t4)

Compute the derivatives and of the vector valued function

We compute,