Podcast
Questions and Answers
Which trigonometric identity is equivalent to $\cos^2(x/2) - \sin^2(x/2)$?
Which trigonometric identity is equivalent to $\cos^2(x/2) - \sin^2(x/2)$?
- $\tan(x)$
- $\cos(x)$ (correct)
- $\sin(x)$
- $\cot(x)$
The expression $\cos(x/2) + \sin(x/2)$ can be simplified to 1.
The expression $\cos(x/2) + \sin(x/2)$ can be simplified to 1.
False (B)
Solve the differential equation: $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ for $\frac{dy}{dx}$.
Solve the differential equation: $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ for $\frac{dy}{dx}$.
$\frac{dy}{dx} = -\frac{2x+y}{x+2y}$
The expression $\cos^2(x/2) + \sin^2(x/2)$ simplifies to ____ .
The expression $\cos^2(x/2) + \sin^2(x/2)$ simplifies to ____ .
Match the following expressions with their simplified forms:
Match the following expressions with their simplified forms:
Which expression represents $1 + 2\sin(x/2)\cos(x/2)$?
Which expression represents $1 + 2\sin(x/2)\cos(x/2)$?
The derivative $\frac{dy}{dx}$ of the equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ is always positive.
The derivative $\frac{dy}{dx}$ of the equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ is always positive.
If $\frac{dy}{dx} = -\frac{2x+y}{x+2y}$, what condition must be satisfied for $\frac{dy}{dx} = 0$?
If $\frac{dy}{dx} = -\frac{2x+y}{x+2y}$, what condition must be satisfied for $\frac{dy}{dx} = 0$?
The differential equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ is a first-order ______ differential equation.
The differential equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ is a first-order ______ differential equation.
Match the following trigonometric expressions with their equivalent forms:
Match the following trigonometric expressions with their equivalent forms:
Given the differential equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$, what is $\frac{dy}{dx}$ when $x = 1$ and $y = -1$?
Given the differential equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$, what is $\frac{dy}{dx}$ when $x = 1$ and $y = -1$?
The equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ is linear.
The equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ is linear.
If $\cos(x/2) = a$ and $\sin(x/2) = b$, express $\cos(x)$ in terms of $a$ and $b$.
If $\cos(x/2) = a$ and $\sin(x/2) = b$, express $\cos(x)$ in terms of $a$ and $b$.
The double angle formula states that $\sin(2x) = 2\sin(x)\cos(x)$. Thus, $\sin(x) = $ ______.
The double angle formula states that $\sin(2x) = 2\sin(x)\cos(x)$. Thus, $\sin(x) = $ ______.
Match the following differential equations with their corresponding derivatives $\frac{dy}{dx}$:
Match the following differential equations with their corresponding derivatives $\frac{dy}{dx}$:
What expression is equivalent to $(\cos(x/2) - \sin(x/2))(\cos(x/2) + \sin(x/2))$?
What expression is equivalent to $(\cos(x/2) - \sin(x/2))(\cos(x/2) + \sin(x/2))$?
If $\frac{dy}{dx} = -\frac{2x+y}{x+2y}$, then $\frac{dx}{dy} = \frac{x+2y}{2x+y}$.
If $\frac{dy}{dx} = -\frac{2x+y}{x+2y}$, then $\frac{dx}{dy} = \frac{x+2y}{2x+y}$.
Express $(\cos(x/2) + \sin(x/2))^2$ in terms of $\sin(x)$ and a constant.
Express $(\cos(x/2) + \sin(x/2))^2$ in terms of $\sin(x)$ and a constant.
To find the derivative $\frac{dy}{dx}$ implicitly for the equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$, we differentiate both sides with respect to ______.
To find the derivative $\frac{dy}{dx}$ implicitly for the equation $2x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$, we differentiate both sides with respect to ______.
Match the following trigonometric identities with their simplified forms:
Match the following trigonometric identities with their simplified forms:
Flashcards
What is cos(x)?
What is cos(x)?
cos²(x/2) - sin²(x/2)
Solve for dy/dx
Solve for dy/dx
dy/dx = -(2x+y)/(x+2y)
Study Notes
- An increase in fluid's speed happens simultaneously with a decrease in pressure or potential energy, based on Bernoulli's principle.
- Bernoulli's principle is expressed as: $P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}$, where:
- $P$ represents Pressure.
- $\rho$ is Density.
- $v$ is Velocity.
- $h$ stands for Height.
- Air moves faster over the top of a wing, which lowers pressure and generates lift.
- Faster moving air corresponds to lower pressure.
- Slower moving air is associated with higher pressure.
Venturi Effect
- The Venturi effect refers to the reduction in fluid pressure when a fluid goes through a constricted section of a pipe.
- When air enters a narrower pipe, its speed increases, which decreases pressure, in turn causing liquid to rise and mix with the airflow.
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