Mathematics, 03.08.2019 22:30 ricky67

Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5). x2 + y2 = (2x2 + 2y2 - x)2

Answers: 1

Show answers

Answers

The equation of the tangent line is $y = x + 0.5$ ====== $x^2 + y^2 = (2x^2+2y^2-x)^2$ the derivative at a point is the slope of the tangent line to that point.to implicitly differentiate, we assume that $y$ is a function of $x$ (so we need to apply chain rule when differentiating y terms)differentiate both sides $\dfrac{\text{d}}{\text{d}x}\left(x^2 + y^2\right) = \dfrac{\text{d}}{\text{d}x}\left(2x^2+2y^2-x\right)^2$ for the left-hand side, use sum rule for differentiation: differentiate each term $\begin{aligned} \dfrac{\text{d}}{\text{d}x}\left(x^2\right) + \dfrac{\text{d}}{\text{d}x}\left(y^2\right) & = \dfrac{\text{d}}{\text{d}x}\left(2x^2+2y^2-x\right)^2 \\ 2x + 2y \tfrac{\text{d}y}{\text{d}x}& = \dfrac{\text{d}}{\text{d}x}\left(2x^2+2y^2-x\right)^2 \end{aligned}$ for the right-hand side, we need to use the chain rule.i will deal with right-hand side separately here, and then add it back to original equation after $\begin{aligned} \text{rhs} & = \dfrac{\text{d}}{\text{d}x}\left(2x^2+2y^2-x\right)^2 \\ & = 2\left(2x^2+2y^2-x\right)^1 \cdot \dfrac{\text{d}}{\text{d}x}\left(2x^2+2y^2-x\right) \\ & = 2\left(2x^2+2y^2-x\right)\left(4x+4y\tfrac{\text{d}y}{\text{d}x}-1\right) \end{aligned}$ let's just connect this together now $\begin{aligned} 2x + 2y \tfrac{\text{d}y}{\text{d}x} & = 2\left(2x^2+2y^2-x\right)\left(4x+4y\tfrac{\text{d}y}{\text{d}x}-1\right) \end{aligned}$ we do not want to simplify this. you can if you want, but it involves a lot of time and unnecessary algebra. you might make mistakes. i might go over the character limit for these answers (around 5000 characters). stuff like that. we do not need to solve for $\text{d}y/\text{d}x$ in terms of $x$ and $y$ when we want to find the derivative at a point for an implicit curve.since we are interested in the point $(0,0.5)$ , substitute in 0 for all $x$ and 0.5 for all $y$ . then solve for $\text{d}y/\text{d}x$ , which is the slope of the tangent line.that x-value of 0 will really us out. $\begin{aligned} 2(0) + 2(0.5) \tfrac{\text{d}y}{\text{d}x} & = 2\left(2(0)^2+2(0.5)^2-0\right)\left(4(0)+4(0.5)\tfrac{\text{d}y}{\text{d}x}-1\right) \\ 1 \tfrac{\text{d}y}{\text{d}x} & = 2\big(2(0.25)\big)\left(2\tfrac{\text{d}y}{\text{d}x}-1\right) \\ \tfrac{\text{d}y}{\text{d}x} & = 2(0.5)\left(2\tfrac{\text{d}y}{\text{d}x}-1\right) \\ \tfrac{\text{d}y}{\text{d}x} & = 2\tfrac{\text{d}y}{\text{d}x}-1 \\ 0 & = \tfrac{\text{d}y}{\text{d}x} - 1 \\ 1 & = \tfrac{\text{d}y}{\text{d}x} \end{aligned}$ since $\text{d}y/\text{d}x$ is 1 at the point $(0,0.5)$ , the slope of the tangent line at point $(0,0.5)$ is 1.therefore, the slope of the line using point-slope form is $\begin{aligned} y - y_1 & = m(x-x_1) \\ y - 0.5 & = 1(x-0) \\ y & = x + 0.5 \end{aligned}$

Answer from: Quest

what is

step-by-step explanation:

Answer from: Quest

Answer: d. 39 cmin the diagram, the approximate length of the minor arc de is 39 centimetres.step-by-step explanation: the formula for the length of an arc in a circle is as displayed below.let the length of the minor arc de = xx = 2( pi )r × angle of sector / 360°from this, we will substitute the values given from the problem into the formula to obtain the length of the minor arc de.x = 2( pi )r × angle of sector / 360°r = radius = 25cmpi as an approximate value = 3.14angle of sector = 120°x = 2( 3.14 )( 25 ) × 120° / 360°x = 157 × 1 / 4x = 39.25x = 39 ( rounded to the nearest whole number as an approximate value )
Ineed as soon as possible ! what is the approximate length of the minor arc de

Ineed as soon as possible ! what is the approximate length of the minor arc de

Another question on Mathematics

Mathematics, 21.06.2019 13:00

What is the equation of the graph that represents f(x)=x^4 stretched vertically by 2 and shifted up 3 spaces

Answers: 1

Answer

Mathematics, 21.06.2019 18:30

Abouncy ball is released from a certain height. on the first bounce, it travels 32cm. with each subsequent bounce, it travels 9/10 the distance of the previous bounce. what is the total distance that the ball has traveled when it completes its 25th bounce?

Answers: 1

Answer

Mathematics, 21.06.2019 19:10

Which ordered pair is a solution of the equation? y + 5 = 2(2+1) choose 1 answer ® only (5,10 ® only (-1,-5) © both (5, 10) and (-1,-5) 0 neither

Answers: 1

Answer

Mathematics, 22.06.2019 00:00

Solve 13 + w/7= -18 (w/7 is a fraction)

Answers: 2

Answer

You know the right answer?

Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0...