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Chapter 3: Problem 29

Find \(d p / d q\). $$p=\frac{\sin q+\cos q}{\cos q}$$

Short Answer

Expert verified
\( \frac{d p}{d q} = \sec^2 q \)

Step by step solution

01

Simplify the Expression

First, simplify the expression given: \[ p = \frac{\sin q + \cos q}{\cos q} \]We can split the fraction into two separate terms:\[ p = \frac{\sin q}{\cos q} + \frac{\cos q}{\cos q} \]This simplifies to:\[ p = \tan q + 1 \]
02

Differentiate "p" with respect to "q"

Next, differentiate the expression for \( p = \tan q + 1 \) with respect to \( q \).The derivative of \( \tan q \) with respect to \( q \) is \( \sec^2 q \), and the derivative of a constant (1) is 0.Thus, the derivative is:\[ \frac{d p}{d q} = \sec^2 q \]

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is an essential task in mathematics that helps you work with equations more easily. Often, an expression looks complicated but can be broken down for simplicity. For example, consider the expression given in the exercise: \[ p = \frac{\sin q + \cos q}{\cos q} \]To simplify, you separate it into two parts:
  • \( \frac{\sin q}{\cos q} \) which is equal to \( \tan q \)
  • \( \frac{\cos q}{\cos q} \) which simplifies to 1
Combining these results, you get:\[ p = \tan q + 1 \]The simplification process is vital because it translates a complex problem into a simpler one, making calculations, like differentiation, easier and less error-prone.
Trigonometric Identities
Trigonometric identities are basic tools in mathematics that allow you to simplify trigonometric functions and solve equations more easily. They are true for all values in their domain and can significantly simplify expressions. Common identities include:
  • \( \sin^2 q + \cos^2 q = 1 \)
  • \( \tan q = \frac{\sin q}{\cos q} \)
  • For the exercise, \( \cos q \cdot \frac{1}{\cos q} = 1 \)
These identities help you transform expressions such as \( \frac{\sin q + \cos q}{\cos q} \) into simpler, easier-to-manage forms like \( \tan q + 1 \). By understanding and using these identities, you can effectively solve equations and find derivatives more efficiently.
Derivatives
Derivatives represent how a function changes as its input changes. It is a central concept in calculus, used extensively for many applications such as calculating rates of change or optimizing functions. In our example, we need to differentiate the expression:\[ p = \tan q + 1 \]Differentiating \( \tan q \) with respect to \( q \) gives \( \sec^2 q \). These steps include:
  • The derivative of \( \tan q \) is \( \sec^2 q \), an essential derivative in calculus.
  • The derivative of a constant like 1 is always 0 because constants don't change.
Hence, the derivative \( \frac{d p}{d q} = \sec^2 q \) emerges. Understanding derivatives helps in exploring how a quantity evolves, which has crucial applications in fields like physics, engineering, and economics.

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Most popular questions from this chapter

Give the position function \(s=f(t)\) of an object moving along the \(s\) -axis as a function of time \(t\). Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t) .\) Comment on the object's behavior in relation to the signs and values of \(v\) and \(a .\) Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? $$s=t^{2}-3 t+2, \quad 0 \leq t \leq 5$$

Find \(d y / d t\) $$y=\sqrt{3 t+\sqrt{2+\sqrt{1-t}}}$$

Show that if it is possible to draw three normals from the point \((a, 0)\) to the parabola \(x=y^{2}\) shown in the accompanying diagram, then \(a\) must be greater than \(1 / 2\). One of the normals is the \(x\) -axis. For what value of \(a\) are the other two normals perpendicular?

a. Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with the properties: i) \(Q(a)=f(a)\) ii) \(Q^{\prime}(a)=f^{\prime}(a)\) iii) \(Q^{\prime \prime}(a)=f^{\prime \prime}(a)\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) b. Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0\) c. Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point (0,1) Comment on what you see. d. Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) Graph \(g\) and its quadratic approximation together. Comment on what you see. e. Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. f. What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts (b), (d), and (e)?

Find \(y^{\prime \prime}\) $$y=x^{2}\left(x^{3}-1\right)^{5}$$
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