2.4 Further Calculus

• What is the parameter commonly denoted as in parametric differentiation?

  t
• What is the key definition of implicit differentiation?

  Differentiate an equation involving both x and y
• When is implicit differentiation used?

  When y is not explicitly in terms of x
• What are the two main advanced integration techniques beyond the basic power rule?

  Substitution and Parts
• What is the key formula used in integration by parts?

  $∫u dv =$$uv - ∫v du$
• Match the integration technique with its definition:

  Integration by Substitution ↔️ Substituting $u =$$g(x)$ to simplify the integral

  Integration by Parts ↔️ Applying the formula $∫u dv =$$uv - ∫v du$
• Match the differentiation technique with its definition:

  Implicit Differentiation ↔️ Differentiation of an implicit equation in x and y

  Parametric Differentiation ↔️ Differentiation of x and y in terms of a parameter t
• What is the formula for the equation of a tangent to a curve?

  $y - y_{1} =$$m(x - x_{1})$
• Integration by Parts is used when the integrand is a product of two functions.


  True
• Separable differential equations are solved by separating variables and integrating.


  True
• What method is used to solve a linear first-order differential equation?

  Integrating factor
• Order the different types of series from simplest to most complex in terms of their formulas:

  1️⃣ Arithmetic Series

  2️⃣ Geometric Series

  3️⃣ Taylor Series
• What is the key characteristic of a Maclaurin series compared to a Taylor series?

  Centered at zero
• In parametric differentiation, both x and y are expressed in terms of a third variable called a parameter
• What is the equation of a line touching a curve at a point?

  $y - y_{1} =$$m(x - x_{1})$
• The slope of a normal line is -\frac{1}{m}</latex>
• Match the integration technique with its description:

  Integration by Substitution ↔️ Simplifying the integral by substituting $u =$$g(x)$

  Integration by Parts ↔️ Applying the formula $\int u \, dv =$$uv - \int v \, du$
• In Integration by Parts, the formula is \int u \, dv = uv - \int v \, du</latex>
• What does the method of volumes of revolution calculate?

  Volume of a 3D shape
• Match the type of first-order differential equation with its solving method:

  Separable ↔️ Separate variables and integrate

  Homogeneous ↔️ Substitute $v =$$\frac{y}{x}$

  Linear ↔️ Use integrating factor
• What is the formula for the sum of an arithmetic series?

  S_{n} = \frac{n}{2}(a_{1} + a_{n})</latex>
• What does the Taylor Series represent?

  A function as an infinite sum
• The formula for the Newton-Raphson method is $x_{n + 1} =$$x_{n} - \frac{f(x_{n})}{f'(x_{n})}$, where $x_{0}$ is the initial guess
• Parametric differentiation allows differentiation with respect to a parameter rather than directly with respect to x or y.


  True
• Parametric differentiation differentiates parametric equations separately with respect to the parameter.


  True
• An example of implicit differentiation is the equation x^{2} + y^{2} = 9
• An example of parametric differentiation is x = 2t, y = 3t^{2</latex>
• Match the integration technique with its description:

  Integration by Substitution ↔️ Simplifies integrands with composite functions

  Integration by Parts ↔️ Applies when integrand is a product of two functions
• What form must the integrand take for Integration by Substitution to be used effectively?

  $f(g(x))g'(x)$
• Implicit Differentiation is used when the dependent variable y is not explicitly in terms of x
• Stationary points occur where the gradient is zero, i.e., $\frac{dy}{dx} =$$0$.


  True
• What is the purpose of finding stationary points in differentiation?

  Find local extrema
• Match the type of first-order differential equation with its description:

  Separable ↔️ Can be written in the form $\frac{dy}{dx} =$$g(x)h(y)$

  Homogeneous ↔️ Can be written in the form $\frac{dy}{dx} =$$f(\frac{y}{x})$

  Linear ↔️ Can be written in the form $\frac{dy}{dx} +$$P(x)y =$$Q(x)$
• What type of first-order differential equation is solved by separating variables and integrating?

  Separable
• What is the general form of a first-order differential equation?

  $\frac{dy}{dx} =$$f(x, y)$
• The formula for the sum of a geometric series is $S_{n} =$$\frac{a(1 - r^{n})}{1 - r}$
• Implicit differentiation is used when y is expressed explicitly in terms of x

  False
• What is the formula for the equation of a normal to a curve at a point?

  $y - y_{1} =$$- \frac{1}{m}(x - x_{1})$
• The equation of a normal line is perpendicular to the tangent at the same point.


  True
• The optimization process involves finding stationary points and using the second derivative test