The Curvature of the Curve Given by the Vector Function $r$

question 36

Multiple Choice

The curvature of the curve given by the vector function $r$ is
$$\mathrm { k } ( t ) = \frac { \left| \mathbf { r } ^ { t } ( t ) \times \mathbf { r } ^ { tt } ( t ) \right| } { \left| \mathbf { r } ^ { t } ( t ) \right| ^ { 3 } }$$
Use the formula to find the curvature of $\mathbf { r } ( t ) = \left\langle \sqrt { 13 } t , e ^ { t } , e ^ { - t } \right)$ at the point $( 0,1,1 )$ .
Select the correct answer.

• A) \(\sqrt { 15 }\)
• B) \(\frac { \sqrt { 2 } } { 15 }\)
• C) \(15 \sqrt { 2 }\)
• D) \(\frac { 15 } { \sqrt { 2 } }\)
• E) \(\frac { \sqrt { 15 } } { 15 }\)

---

Definitions:

Present Value

Present value is the current worth of a future sum of money or stream of cash flows given a specified rate of return.

Compounded Quarterly

A method of calculating interest where the interest is added to the principal amount every quarter, leading to interest being earned on interest.

Compounded Semiannually

Refers to the process where interest on a financial product, like a savings account, is calculated and added to the principal amount twice a year.

Compounded Quarterly

The process where interest is calculated and added to the principal sum of an investment or loan on a quarterly basis, leading to an increase in the amount of interest earned over time.