3.5 Calculus (Higher Tier)

• What does differentiation describe in calculus?

  Gradient of a curve
• Differentiation is crucial in physics, engineering, and economics
• Differentiation finds the slope or gradient of a curve at a given point.


  True
• Match the concept with its definition:

  Differentiation ↔️ Finding the derivative of a function

  Gradient ↔️ The slope of a curve

  Rate of Change ↔️ How quickly a function is changing
• The derivative of a function describes its rate of change
• The derivative of a function represents the gradient or slope of the curve at a given point.


  True
• Match the concept with its definition:

  Differentiation ↔️ Finding the derivative of a function

  Gradient ↔️ The slope of a curve

  Rate of Change ↔️ How quickly a function is changing
• The Power Rule states that the derivative of $x^{n}$ is $nx^{n - 1}$, where you reduce the exponent by 1.
• What is the derivative of $x^{5}$ using the Power Rule?


  $5x^{4}$
• The Constant Multiple Rule states that the derivative of $kf(x)$ is $k\frac{d}{dx}f(x)$.


  True
• What is the derivative of $3x^{2}$ using the Constant Multiple Rule?


  $6x$
• What does the Constant Multiple Rule state about differentiating $kf(x)$?


  $k\frac{d}{dx}f(x)$
• The derivative of $x^{5}$ using the Power Rule is $5x^{4}$


  True
• What is the derivative of $3x^{2}$ using the Constant Multiple Rule?


  $6x$
• The derivative of $2^{x}$ using the Exponential Rule is $2^{x} \ln(2)$


  True
• What is the derivative of 4x^{3} + 2x^{2} - x + 6</latex>?

  $12x^{2} +$$4x - 1$
• What does differentiation describe in calculus?

  Gradient of a curve
• Differentiation represents the rate of change of a function at a specific point.

  True
• What does the Power Rule state for the derivative of $x^{n}$?


  $nx^{n - 1}$
• Using the Power Rule to differentiate $f(x) =$$x^{5}$, the result is $5x^{4}$
• What does the Constant Multiple Rule state for the derivative of $kf(x)$?


  k\frac{d}{dx}f(x)</latex>
• Using the Constant Multiple Rule to differentiate $f(x) =$$3x^{2}$, the result is $6x$
• Steps to apply the Power Rule and Constant Multiple Rule

  1️⃣ Identify the function and its type

  2️⃣ Apply the Power Rule if necessary

  3️⃣ Apply the Constant Multiple Rule if necessary

  4️⃣ Simplify the result
• The Power Rule involves multiplying the original exponent by the coefficient and reducing the exponent by 1.


  True
• What is the derivative of $x^{ - 3}$ using the Power Rule?


  $- 3x^{ - 4}$
• When differentiating $f(x) =$$4x^{4}$, the result is 16x^{3}</latex>
• What is the derivative of $f(x) =$$3x^{3} - 2x +$$7$?


  $9x^{2} - 2$
• Polynomials are expressions with variables raised to positive integer powers multiplied by coefficients.

  True
• While differentiation finds the rate of change, integration finds the antiderivative
• Match the calculus concept with its objective:

  Differentiation ↔️ Determines the slope or gradient

  Integration ↔️ Finds the area under the curve
• What is the Power Rule for Integration?

  \int x^{n} dx = \frac{x^{n + 1}}{n + 1} + C</latex>
• What does the Power Rule for Integration allow us to find?

  The integral of $x^{n}$
• The indefinite integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1} +$$C$
• The exponent $n$ in the Power Rule cannot be equal to -1.


  True
• What does the term $C$ represent in the Power Rule?


  Constant of integration
• Steps to apply the Power Rule for Integration

  1️⃣ Increase the exponent by 1: $n +$$1$

  2️⃣ Divide by the new exponent: $\frac{1}{n + 1}$

  3️⃣ Add the constant of integration: $+$$C$
• What is the integral of f(x) = x^{3}</latex> using the Power Rule?

  $\frac{x^{4}}{4} +$$C$
• The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1} +$$C$
• What is the general form of an exponential function?

  $a^{x}$
• The Exponential Rule states that the derivative of $a^{x}$ is $a^{x} \ln(a)$