007A Sample Midterm 3, Problem 4 Detailed Solution

Consider the circle  $x^{2}+y^{2}=25.$

(a)  Find  ${\frac {dy}{dx}}.$

(b)  Find the equation of the tangent line at the point  $(4,-3).$

---

Background Information:
1. What is the result of implicit differentiation of  $y^{2}?$
It would be  $2y\cdot {\frac {dy}{dx}}$  by the Chain Rule.
2. 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.
3. What is the slope of the tangent line of a curve?
The slope is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m={\frac {dy}{dx}}.}

Solution:

(a)

Step 1:
Using implicit differentiation on the equation  $x^{2}+y^{2}=25,$  we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2x+2y\cdot {\frac {dy}{dx}}=0.}

Step 2:
Now, solve for  ${\frac {dy}{dx}}.$
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 2y\cdot\frac{dy}{dx}=-2x.}
We solve to get
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}=-\frac{x}{y}.}

(b)

Step 1:
First, we find the slope of the tangent line at the point  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 (4,-3).}
We plug  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 (4,-3)}   into the formula for  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}}   we found in part (a).
So, we get
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 \begin{array}{rcl} \displaystyle{m} & = & \displaystyle{-\bigg(\frac{4}{-3}\bigg)}\\ &&\\ & = & \displaystyle{\frac{4}{3}.} \end{array}}

Step 2:
Now, we have the slope of the tangent line at  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 (4,-3)}   and a point.
Thus, we can write the equation of the line.
So, the equation of the tangent line at  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 (4,-3)}   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\,=\,\frac{4}{3}(x-4)-3.}

Final Answer:
(a)    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}=-\frac{x}{y}}
(b)    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\,=\,\frac{4}{3}(x-4)-3}

Return to Sample Exam