009A Sample Final 1, Problem 4

If

$y=x^{2}+\cos(\pi (x^{2}+1))$

compute  ${\frac {dy}{dx}}$  and find the equation for the tangent line at $x_{0}=1$. You may leave your answers in point-slope form.

Foundations:
1. What two pieces of information do you need to write the equation of a line?
You need the slope of the line and a point on the line.
2. What does the Chain Rule state?
For functions $f(x)$  and $g(x),$  $~{\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x).$

Solution:

2

Step 1:
First, we compute ${\frac {dy}{dx}}.$ We get
${\frac {dy}{dx}}=2x-\sin(\pi (x^{2}+1))(2\pi x).$

Step 2:
To find the equation of the tangent line, we first find the slope of the line.
Using $x_{0}=1$ in the formula for ${\frac {dy}{dx}}$ from Step 1, we get
$m=2(1)-\sin(2\pi )2\pi =2.$
To get a point on the line, we plug in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0=1} into the equation given.
So, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=1^2+\cos(2\pi)=2.}
Thus, the equation of the tangent line is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=2(x-1)+2.}

1

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=2(x-1)+2}

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