What Did I Do Wrong? Analyzing y = x tan x

  • Thread starter PrudensOptimus
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In summary, the purpose of analyzing y = x tan x is to understand its properties and behavior. To graph it, we can plot key points and use a graphing calculator. The key properties include its domain, range, and asymptotes. Equations involving y = x tan x can be solved using algebraic techniques, but we must check for extraneous solutions. This function has various real-world applications, such as calculating heights and angles, and is useful in fields such as engineering and physics.
  • #1
PrudensOptimus
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y = x tan x

y' = tan x + x*sec^2(x)

y'' = sec^2(x) + d/dx (x*sec^2(x))

= ... + (d/dx(sec^2(x)) + sec^2(x))

= ... + (2secx + sec^2(x))

= 2sec^3 x + sec^4(x)

what did I do wrong?? in book it says y'' = something else.
 
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  • #2
= ... + (2secx + sec^2(x))
First term should have d/dx(secx)=tanx.secx multipled, i.e. first term is 2sec2x.tanx.
 
  • #3
= ... + (d/dx(sec^2(x)) + sec^2(x))

= ... + (2secx + sec^2(x))

= 2sec^3 x + sec^4(x)

You multiplied the first "sec^2 x" instead of adding!
 
  • #4
I found the answer this morning,

something like

2sec^2 x(tanx + 1)
 

1. What is the purpose of analyzing y = x tan x?

The purpose of analyzing y = x tan x is to understand the behavior and properties of the function. By studying its graph and equation, we can determine key features such as the domain, range, intercepts, and asymptotes. This can help us make predictions and solve practical problems involving the function.

2. How do I graph y = x tan x?

To graph y = x tan x, we can first plot some key points by substituting different values for x, such as 0, 1, -1, π/4, π/2, etc. Then, we can connect these points with a smooth curve, paying attention to any asymptotes or unusual behavior. It is also helpful to use a graphing calculator or software to visualize the graph and make adjustments.

3. What are the key properties of y = x tan x?

The key properties of y = x tan x include:

  • Domain: all real numbers except odd multiples of π/2
  • Range: all real numbers
  • x-intercept: 0
  • y-intercept: 0
  • Asymptotes: x = odd multiples of π/2

4. How can I solve equations involving y = x tan x?

To solve equations involving y = x tan x, we can use algebraic techniques such as factoring, substitution, or trigonometric identities. It is also important to check for extraneous solutions, as the tangent function has periodic behavior and may have multiple solutions for a given x-value.

5. In what real-world situations can y = x tan x be applied?

Y = x tan x can be applied in various real-world situations, such as calculating the height of a flagpole based on its shadow, determining the angle of a ladder against a wall, or analyzing the oscillations of a pendulum. It also has applications in engineering, physics, and other fields where periodic or exponential growth/decay is involved.

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