A curve is defined by the parametric equations $x(t)=10 \cos (9 t)$ and $y(t)=4 e^{10 t}$ . Find $\frac{d y}{d x}$ . $\newline$ Answer:

View step-by-step help

Home

Math Problems

Algebra 1

Write a quadratic function from its x-intercepts and another point

Full solution

Q. A curve is defined by the parametric equations

x(t)=10 \cos (9 t)

and

y(t)=4 e^{10 t}

. Find

\frac{d y}{d x}

\newline

Answer:

Find Derivative of $x(t)$ : To find the derivative $\frac{dy}{dx}$ for the parametric equations, we first need to find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ separately. $\newline$ For $x(t) = 10 \cos(9t)$ , the derivative with respect to $t$ is $\frac{dx}{dt} = -90 \sin(9t)$ .
Find Derivative of $y(t)$ : Now, we find the derivative of $y(t)$ with respect to $t$ . For $y(t) = 4e^{10t}$ , the derivative with respect to $t$ is $\frac{dy}{dt} = 40e^{10t}$ .
Calculate $(\frac{dy}{dx}):</b> The derivative \$(\frac{dy}{dx}) is found by dividing \$\frac{dy}{dt}$ by $\frac{dx}{dt}$ . $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{40e^{10t}}{-90 \sin(9t)}$ .
Simplify the Expression: We simplify the expression for $\frac{dy}{dx}$ . $\frac{dy}{dx} = - \frac{40}{90} \times \frac{e^{10t}}{\sin(9t)} = - \frac{4}{9} \times \frac{e^{10t}}{\sin(9t)}$ .

More problems from Write a quadratic function from its x-intercepts and another point

Question
Solve by completing the square. $\newline$ $m^2 - 10m - 29 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ `m` = ____ or `m` = _____
Get tutor help
Posted 8 months ago

Question
Find $g(x)$ , where $g(x)$ is the translation $5$ units up of $f(x)=x^2$ . $\newline$ Write your answer in the form $a(x–h)^2+k$ , where $a$ , $h$ , and $k$ are integers. $\newline$ $g(x)=$ ____
Get tutor help
Posted 7 months ago

Question
What is the range of this quadratic function? $\newline$ $y = x^2 - 4x + 4$ $\newline$ Choices: $\newline$ $\left\{y \mid y \geq 2\right\}$ $\newline$ $\left\{y \mid y \leq 0\right\}$ $\newline$ $\left\{y \mid y \geq 0\right\}$ $\newline$ $\text{all real numbers}$
Get tutor help
Posted 5 months ago

Question
Write the equation of the parabola that passes through the points $(1,0)$ , $(2,0)$ , and $(3,\text{–}16)$ . Write your answer in the form $y = a(x – p)(x – q)$ , where $a$ , $p$ , and $q$ are integers, decimals, or simplified fractions. $\newline$ ______
Get tutor help
Posted 5 months ago

Question
Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic. $\newline$ $f^2 + 8f + \_\_\_\_\_$
Get tutor help
Posted 5 months ago

Question
Solve for $h$ . $\newline$ $h^2 + 39h = 0$ $\newline$ $\newline$ Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas. $\newline$ $h =$ ____ $\newline$
Get tutor help
Posted 5 months ago

Question
Write a quadratic function with zeros $-9$ and $-7$ . $\newline$ Write your answer using the variable $x$ and in standard form with a leading coefficient of $1$ . $\newline$ $f(x) = \_\_\_\_\_$
Get tutor help
Posted 5 months ago

Question
Find the equation of the axis of symmetry for the parabola $y = x^2$ . $\newline$ Simplify any numbers and write them as proper fractions, improper fractions, or integers. $\newline$ $\underline{\hspace{3cm}}$
Get tutor help
Posted 4 months ago

Question
Find $g(x)$ , where $g(x)$ is the translation $8$ units up of $f(x) = x^2$ . $\newline$ Write your answer in the form $a(x – h)^2 + k$ , where $a$ , $h$ , and $k$ are integers. $\newline$ $g(x) =$ ______ $\newline$
Get tutor help
Posted 4 months ago

Question
Solve for $x$ . $\newline$ $x^2 = 1$ $\newline$ $\newline$ Write your answer in simplified, rationalized form. $\newline$ $x =$ ______ or $x =$ ______ $\newline$
Get tutor help
Posted 5 months ago

View all (106)

A curve is defined by the parametric equations x(t)=10cos⁡(9t) x(t)=10 \cos (9 t) x(t)=10cos(9t) and y(t)=4e10t y(t)=4 e^{10 t} y(t)=4e10t. Find dydx \frac{d y}{d x} dxdy​.\newlineAnswer:

A curve is defined by the parametric equations $x(t)=10 \cos (9 t)$ and $y(t)=4 e^{10 t}$ . Find $\frac{d y}{d x}$ . $\newline$ Answer: