Problem 86 - University Calculus: Early Transcendentals

Question

If \(r=\sin (f(t)), f(0)=\pi / 3,\) and \(f^{\prime}(0)=4,\) then what is \(d r / d t\) at \(t=0 ?\)

Step-by-Step Solution

Answer

The derivative \( \frac{dr}{dt} \) at \( t=0 \) is 2.

Step 1: Identify the Given Information

We are given that \( r = \sin(f(t)), \) \( f(0) = \frac{\pi}{3}, \) and \( f'(0) = 4. \) We need to find the derivative \( \frac{dr}{dt} \) at \( t = 0. \)

Step 2: Apply the Chain Rule

Since \( r = \sin(f(t)) \), to find \( \frac{dr}{dt} \), we use the chain rule: \[ \frac{dr}{dt} = \cos(f(t)) \cdot \frac{df}{dt}. \]

Step 3: Evaluate Each Part at \( t = 0 \)

To find \( \frac{dr}{dt} \) at \( t = 0 \), substitute \( t = 0 \) into the derivative expression. Using the given \( f(0) = \frac{\pi}{3} \), we have \( \cos(f(0)) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}. \) We are also given \( f'(0) = 4. \)

Step 4: Calculate \(\frac{dr}{dt}\) at \(t=0\)

Combine the parts: \[ \frac{dr}{dt} = \cos(f(0)) \cdot f'(0) = \frac{1}{2} \cdot 4 = 2. \] Thus, \( \frac{dr}{dt} = 2 \) at \( t = 0. \)

Key Concepts

Understanding Derivatives

Derivatives represent the rate at which a function changes as its input changes. In simple terms, they tell us how one quantity changes in relation to another. When applied to motion, for example, a derivative can represent velocity, showing us how position changes over time.

In calculus, finding a derivative usually involves applying specific rules to differentiate a function. These rules, like the power rule or the product rule, make it easier to work with various types of functions. When dealing with more complex functions like those involving trigonometric functions or when chaining functions together, the chain rule becomes crucial to finding the derivative.

Trigonometric Functions and Their Derivatives

Trigonometric functions are fundamental in calculus and appear often in physics, engineering, and other fields. The basic trigonometric functions include sine (\(\sin(x)\)), cosine (\(\cos(x)\)), and tangent (\(\tan(x)\)). Each of these functions has specific derivatives that we should remember:

• The derivative of \(\sin(x)\) is \(\cos(x)\).
• The derivative of \(\cos(x)\) is \(-\sin(x)\).
• The derivative of \(\tan(x)\) is \(\sec^2(x)\).

Understanding these helps in calculating the derivative of more complicated expressions that include trigonometric functions. For example, in the given problem, we used the derivative of the sine function to help determine the rate of change of \(r(t)\).

Exploring Differentiation Techniques

Differentiation techniques are methods used to find the derivative of functions that are not directly differentiated through basic rules. One of the most important techniques is the **chain rule**. The chain rule applies when you have a "function within a function" situation, or a composition of functions. It states that if you have a compound function \(g(x) = f(h(x))\), the derivative \(g'(x)\) is found by:

• First differentiating the outer function \(f(x)\) with respect to the inner function \(h(x)\).
• Then differentiating the inner function \(h(x)\) with respect to \(x\).
• Finally, multiplying these two derivatives together.

In the context of the exercise, the chain rule allows us to differentiate \(r = \sin(f(t))\) with respect to \(t\) by considering how \(r\) changes not only with \(f(t)\), but also with how \(f(t)\) changes with \(t\), giving us a full picture of its rate of change.