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Chapter 3: Problem 19

In Exercises \(19-22,\) find \(d s / d t\) $$ s=\tan t-t $$

Short Answer

Expert verified
\( \frac{d s}{d t} = \sec^2 t - 1 \)

Step by step solution

01

Differentiate each term with respect to t

To find \( \frac{d s}{d t} \), we need to differentiate each term of the given function. The function is \( s = \tan t - t \). Differentiate \( \tan t \) with respect to \( t \), which gives \( \sec^2 t \), and differentiate \( t \) with respect to \( t \), which gives \( 1 \).
02

Combine the derivatives

Now, combine the differentiated terms: \( \frac{d}{d t} (\tan t) = \sec^2 t \) and \( \frac{d}{d t} (-t) = -1 \). Therefore, \( \frac{d s}{d t} = \sec^2 t - 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of Trigonometric Functions
When finding the derivative of trigonometric functions, it's crucial to remember the basic rules and identities associated with them. In calculus, understanding how to differentiate trigonometric functions such as sine, cosine, and tangent is essential.
For this exercise, the function given is \(s = \tan t - t\). To differentiate \(\tan t\), we use the identity that the derivative of \(\tan t\) is \(\sec^2 t\). This is derived from using the quotient rule and the fundamental trigonometric identities.
Each trigonometric function has its own derivative:
  • The derivative of \(\sin t\) is \(\cos t\).
  • The derivative of \(\cos t\) is \(-\sin t\).
  • The derivative of \(\tan t\) is \(\sec^2 t\).
Understanding these derivatives allows us to solve more complex calculus problems involving trigonometric functions.
Calculus Problem Solving
Solving calculus problems often involves breaking down functions into simpler components before applying differentiation rules. The function \(s = \tan t - t\) can be approached by individually differentiating each term.
This process involves:
  • Identifying and simplifying each part of the function where necessary
  • Applying known differentiation rules to each component
  • Combining the results to find the overall derivative
By following these steps systematically, solving complex calculus problems becomes manageable. For this exercise, we identified the derivatives as \(\sec^2 t\) from \(\tan t\) and \(-1\) from \(-t\), leading to the solution \(\frac{d s}{d t} = \sec^2 t - 1\).
Understanding how to dissect and approach each part is key to mastering calculus.
Chain Rule in Calculus
Even though this specific problem does not explicitly require the use of the chain rule, it's important to understand how it applies to more complex differentiations involving trigonometric functions.
The chain rule is applied when differentiating composite functions. If you have a function like \(s(u(t))\), the derivative is found by multiplying the derivative of the outer function \(s\) by the derivative of the inner function \(u(t)\).
In mathematical terms, it's expressed as:
  • \(\frac{d}{dt}[s(u(t))] = s'(u(t)) \cdot u'(t)\)
This rule is particularly useful in calculus when functions involve nested or embedded expressions. It simplifies managing layers within functions and ensures exact and precise differentiation.
Thus, becoming comfortable with the chain rule expands the capability to solve a wider range of calculus problems effectively.

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Most popular questions from this chapter

Estimating volume Estimate the volume of material in a cylindrical shell with length 30 in... radius 6 in. and shell thickness 0.5 in.

In Exercises \(57-60,\) use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval I. Perform the following steps: $$ \begin{array}{l}{\text { a. Plot the function } f \text { over } I} \\ {\text { b. Find the linearization } L \text { of the function at the point } a \text { . }} \\ {\text { c. Plot } f \text { and } L \text { together on a single graph. }} \\ {\text { d. Plot the absolute error }|f(x)-L(x)| \text { over } I \text { and find its max- }} \\ {\text { imum value. }}\end{array} $$ $$ \begin{array}{l}{\text { e. From your graph in part (d), estimate as large a } \delta>0 \text { as you }} \\ {\text { can, satisfing }}\end{array} $$ $$ \begin{array}{c}{|x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\epsilon} \\\ {\text { for } \epsilon=0.5,0.1, \text { and } 0.01 . \text { Then check graphically to see if }} \\ {\text { your } \delta \text { -estimate holds true. }}\end{array} $$ $$ f(x)=\frac{x-1}{4 x^{2}+1}, \quad\left[-\frac{3}{4}, 1\right], \quad a=\frac{1}{2} $$

If \(r+s^{2}+v^{3}=12, d r / d t=4,\) and \(d s / d t=-3,\) find \(d v / d t\) when \(r=3\) and \(s=1.\)

Diagonals If \(x, y,\) and \(z\) are lengths of the edges of a rectangular box, the common length of the box's diagonals is \(s=\) \(\sqrt{x^{2}+y^{2}+z^{2}}\) a. Assuming that \(x, y,\) and \(z\) are differentiable functions of \(t\) how is \(d s / d t\) related to \(d x / d t, d y / d t,\) and \(d z / d t\) ? b. How is \(d s / d t\) related to \(d y / d t\) and \(d z / d t\) if \(x\) is constant? c. How are \(d x / d t, d y / d t,\) and \(d z / d t\) related if \(s\) is constant?

Use a CAS to perform the following steps for the functions \begin{equation} \begin{array}{l}{\text { a. Plot } y=f(x) \text { to see that function's global behavior. }} \\ {\text { b. Define the difference quotient } q \text { at a general point } x, \text { with }} \\ {\text { general step size } h .} \\\ {\text { c. Take the limit as } h \rightarrow 0 . \text { What formula does this give? }} \\ {\text { d. Substitute the value } x=x_{0} \text { and plot the function } y=f(x)} \\ \quad {\text { together with its tangent line at that point. }} \\ {\text { e. Substitute various values for } x \text { larger and smaller than } x_{0} \text { into }} \\ \quad {\text { the formula obtained in part (c). Do the numbers make sense }} \\ \quad {\text { with your picture? }} \\ {\text { f. Graph the formula obtained in part (c). What does it mean }} \\ \quad {\text { when its values are negative? Zero? Positive? Does this make }} \\ \quad {\text { sense with your plot from part (a)? Give reasons for your }} \\ \quad {\text { answer. }}\end{array} \end{equation} $$f(x)=\frac{x-1}{3 x^{2}+1}, \quad x_{0}=-1$$
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