91Ó°ÊÓ

When dealing with vector functions like \( \mathbf{r}(t) \ = 4 \mathbf{i} - \cos t \mathbf{j} \), it's important to understand the concept of derivatives. A derivative represents the rate at which a function changes at any given point. For functions of a single variable, this is straightforward. However, with vector functions, differentiation involves dealing with each component separately.

In this exercise, the function \( \mathbf{r}(t) \) is composed of two components: one in the direction of \( \mathbf{i} \), and another in the direction of \( \mathbf{j} \). We tackle each part independently because vector functions can have multi-dimensional outputs:

• The \( \mathbf{i} \) component, \( 4 \), is a constant, and its derivative is simply \( 0 \), as constants don't change.
• The \( \mathbf{j} \) component, \( -\cos t \), changes over time, requiring us to apply the standard rules of differentiation for trigonometric functions.

By understanding each component's derivative, we can construct the overall derivative of the vector function.

Vectors are crucial in representing quantities that have both magnitude and direction. In a vector function like \( \mathbf{r}(t) \ = 4 \mathbf{i} - \cos t \mathbf{j} \), we break down the vector into its basic components:

• The first component is aligned along the \( \mathbf{i} \)-axis. In our problem, this is the term \( 4 \mathbf{i} \), which is simply a constant vector pointing in the \( \mathbf{i} \) direction.
• The second component is aligned along the \( \mathbf{j} \)-axis. Here, it is given by \( -\cos t \mathbf{j} \), indicating a dynamic element that changes based on the cosine of \( t \).

By analyzing these components, we can easily apply vector differentiation strategies. Each component is treated separately, ensuring a systematic approach to solving problems involving vectors. This breakdown helps in understanding how changes in each direction affect the behavior of the vector as a whole.

Trigonometric functions like \( \cos t \) and \( \sin t \) often appear in calculus problems, including vector differentiation. Knowing their derivatives is key:

The derivative of \( \cos t \) with respect to \( t \) is \( -\sin t \). This is a fundamental rule:

• When we differentiate \( -\cos t \), we need to consider the negative sign. Thus, it becomes \( \sin t \) after differentiation.

Understanding this transformation is crucial, especially when dealing with vector components that incorporate trigonometric functions.

These derivative rules allow us to accurately determine how a function's rate of change evolves over time. By systematically applying these principles, we ensure accurate and reliable results in mathematical computations involving trigonometric derivatives.