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Chapter 3: Problem 22

In Exercises \(19-22,\) find \(d s / d t\) $$ s=\frac{\sin t}{1-\cos t} $$

Short Answer

Expert verified
\(ds/dt = -\frac{1 + \cos t}{(1 - \cos t)^2}\)

Step by step solution

01

Identify the Function Components

The function given is \( s = \frac{\sin t}{1 - \cos t} \). It is a quotient, implying that the quotient rule will be used to find the derivative.
02

Recall and Write the Quotient Rule

The quotient rule for derivatives states that if \( u(t) = \frac{f(t)}{g(t)} \), then \( \frac{d}{dt}[u(t)] = \frac{f'(t)g(t) - f(t)g'(t)}{[g(t)]^2} \). Here, \( f(t) = \sin t \) and \( g(t) = 1 - \cos t \).
03

Find Derivatives of the Numerator and Denominator

Calculate the derivatives of \( f(t) \) and \( g(t) \).- Derivative of \( f(t) = \sin t \) is \( f'(t) = \cos t \).- Derivative of \( g(t) = 1 - \cos t \) is \( g'(t) = \sin t \).
04

Apply the Quotient Rule

Substitute \( f(t), f'(t), g(t), \) and \( g'(t) \) into the quotient rule formula:\[\frac{d}{dt} \left( \frac{\sin t}{1 - \cos t} \right) = \frac{(\cos t)(1 - \cos t) - (\sin t)(\sin t)}{(1 - \cos t)^2}\]
05

Simplify the Expression

Simplify the expression obtained:- Expand \( (\cos t)(1 - \cos t) \): \( \cos t - \cos^2 t \).- Expand \( (\sin t)(\sin t) \) to \( \sin^2 t \).- The numerator becomes \( \cos t - \cos^2 t - \sin^2 t \).
06

Use Trigonometric Identities

Recall that \( \sin^2 t + \cos^2 t = 1 \).- Therefore, \( \cos^2 t = 1 - \sin^2 t \).- Substitute \( \cos^2 t \) in the expression: \( \cos t - (1 - \sin^2 t) - \sin^2 t = \cos t - 1 \).
07

Finalize the Derivative

The final expression is:\[\frac{\cos t - 1 - \sin^2 t}{(1 - \cos t)^2} = -\frac{1 + \cos t}{(1 - \cos t)^2}\]

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Key Concepts

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

Trigonometric Identities
Trigonometric identities play a crucial role in simplifying and solving problems involving trigonometric functions. In the exercise, we use the identity \( \sin^2 t + \cos^2 t = 1 \) to replace \( \cos^2 t \) in our expression. By recognizing this identity, we can reframe trigonometric expressions for easier manipulation. This identity allows us to convert terms involving \( \cos^2 t \) into terms with only \( \sin^2 t \) and constants, facilitating simplification.
  • Remember that \( \sin^2 t + \cos^2 t = 1 \) is always true for any angle \( t \).
  • It helps simplify expressions by reducing the number of terms.
  • Used to derive and verify other trigonometric identities.
Understanding these identities enhances problem-solving skills, as they frequently appear in derivative calculations and other algebraic manipulations.
Derivative Calculation
Calculating derivatives is a fundamental skill in calculus, particularly when dealing with complex functions like the one in this exercise. The quotient rule is a technique used to take derivatives of functions that are ratios of two smaller functions. It is crucial to understand this rule, as it simplifies finding derivatives of fractions:
  • The formula for the quotient rule is \( \frac{d}{dt}\left[\frac{f(t)}{g(t)}\right]=\frac{f'(t)g(t)-f(t)g'(t)}{[g(t)]^2} \).
  • Apply the derivatives of the numerator \( f(t) \) and the denominator \( g(t) \).
  • Substitute these into the quotient rule formula to find the derivative.
By practicing the quotient rule, you will become more comfortable with differentiating complex rational functions, effectively breaking down problems into simpler components.
Simplifying Expressions
After finding a derivative, the next step is simplification, which is often where problems become manageable. Simplification in calculus involves using algebraic and trigonometric techniques to transform the derivative into its most concise form.In our example, we initially obtain a complicated expression after applying the quotient rule. Simplifying involves expanding terms like \( (\cos t)(1 - \cos t) \) and reducing them into simpler components by adding or subtracting similar terms:
  • Identify opportunities to combine like terms.
  • Use trigonometric identities to substitute and simplify expressions further, such as replacing \( \cos^2 t \) with \( 1 - \sin^2 t \).
  • Structure the final expression in a more manageable form, such as \( -\frac{1 + \cos t}{(1 - \cos t)^2} \) in this problem.
Mastering simplification ensures clarity and precision in mathematical communication, making it easier to interpret the final result.

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Most popular questions from this chapter

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} $$f(x)=x^{2} \cos x, \quad x_{0}=\pi / 4$$

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: $$ \begin{array}{l}{\text { a. Plot the function } f \text { over } I} \\ {\text { b. Find the linearization } L \text { of the function at the point } a \text { . }} \\ {\text { c. Plot } f \text { and } L \text { together on a single graph. }} \\ {\text { d. Plot the absolute error }|f(x)-L(x)| \text { over } I \text { and find its max- }} \\ {\text { imum value. }}\end{array} $$ $$ \begin{array}{l}{\text { e. From your graph in part (d), estimate as large a } \delta>0 \text { as you }} \\ {\text { can, satisfing }}\end{array} $$ $$ \begin{array}{c}{|x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\epsilon} \\\ {\text { for } \epsilon=0.5,0.1, \text { and } 0.01 . \text { Then check graphically to see if }} \\ {\text { your } \delta \text { -estimate holds true. }}\end{array} $$ $$ f(x)=x^{2 / 3}(x-2), \quad[-2,3], \quad a=2 $$

Cardiac output In the late 1860 \(\mathrm{s}\) , Adolf Fick, a professor of physiology in the Faculty of Medicine in Wurzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 \(\mathrm{L} / \mathrm{min.}\) At rest it is likely to be a bit under 6 \(\mathrm{L} / \mathrm{min.}\) If you are a trained marathon runner running a marathon, your cardiac output can be as high as 30 \(\mathrm{L} / \mathrm{min} .\) $$\begin{array}{c}{\text { Your cardiac output can be calculated with the formula }} \\ {y=\frac{Q}{D}}\end{array}$$ where \(Q\) is the number of milliliters of \(\mathrm{CO}_{2}\) you exhale in a minute and \(D\) is the difference between the \(\mathrm{CO}_{2}\) concentration \((\mathrm{ml} / \mathrm{L})\) in the blood pumped to the lungs and the \(\mathrm{CO}_{2}\) concentration in the blood returning from the lungs. With \(Q=233 \mathrm{ml} / \mathrm{min}\) and \(D=97-56=41 \mathrm{ml} / \mathrm{L},\) $$y=\frac{233 \mathrm{ml} / \min }{41 \mathrm{ml} / \mathrm{L}} \approx 5.68 \mathrm{L} / \mathrm{min},$$ fairly close to the 6 \(\mathrm{L} / \mathrm{min}\) that most people have at basal (resting conditions. (Data courtesy of J. Kenneth Herd, M.D. Quillan College of Medicine, East Tennessee State University.) Suppose that when \(Q=233\) and \(D=41,\) we also know that \(D\) is decreasing at the rate of 2 units a minute but that \(Q\) remains unchanged. What is happening to the cardiac output?

Distance Let \(x\) and \(y\) be differentiable functions of \(t\) and let \(s=\sqrt{x^{2}+y^{2}}\) be the distance between the points \((x, 0)\) and \((0, y)\) in the \(x y\)-plane. a. How is \(d s / d t\) related to \(d x / d t\) if \(y\) is constant? b. How is \(d s / d t\) related to \(d x / d t\) and \(d y / d t\) if neither \(x\) nor \(y\) is constant? c. How is \(d x / d t\) related to \(d y / d t\) if \(s\) is constant?

If \(L=\sqrt{x^{2}+y^{2}}, d x / d t=-1,\) and \(d y / d t=3,\) find \(d L / d t\) when \(x=5\) and \(y=12 .\)
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