MM 34 Unit Book 2025 (1) (1)

Differentiation:

9A) The Derivative

9B&C) Rules for Differentiation

• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$

• Constant Multiple Rule: If $f(x) = c \cdot g(x)$ , then $f'(x) = c \cdot g'(x)$

• Sum Rule: If $f(x) = u(x) + v(x)$, then $f'(x) = u'(x) + v'(x)$

• Difference Rule: If $f(x) = u(x) - v(x)$, then $f'(x) = u'(x) - v'(x)$

9D) The Graph of the Gradient Function

9E & 9F) The Chain Rule

• If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

9G) Derivative of $e^x$

• If $f(x) = e^x$, then $f'(x) = e^x$

• If $f(x) = e^{kx}$, then $f'(x) = ke^{kx}$

9H) Derivative of $log_e(x)$

• If $f(x) = log_e(x) = ln(x)$, then $f'(x) = \frac{1}{x}$

• If $f(x) = log_e(kx)$, then $f'(x) = \frac{1}{x}$

9I) Derivative of Trigonometric Functions

• If $f(x) = sin(x)$, then $f'(x) = cos(x)$

• If $f(x) = cos(x)$, then $f'(x) = -sin(x)$

• If $f(x) = tan(x)$, then $f'(x) = sec^2(x) = 1 + tan^2(x)$

9J) The Product Rule

• If $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$

9K) The Quotient Rule

• If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

9L & M) Limits and Continuity and Conditions of Differentiability

Applications of Differentiation:

10A) Equations of Tangents and Normals

• Tangent: $y - y<em>1 = m(x - x</em>1)$, where m is the derivative at $(x<em>1, y</em>1)$

• Normal: $y - y<em>1 = -\frac{1}{m}(x - x</em>1)$, where -1/m is the negative reciprocal of the derivative at $(x<em>1, y</em>1)$

10B) Rates of Change

• Related Rates: Use chain rule to relate rates of change of different variables.

10C&D) Stationary Points and Their Nature

• Stationary Points: Points where $f'(x) = 0$

• Nature of Stationary Points:

  • Maximum: f''(x) < 0

  • Minimum: f''(x) > 0

  • Point of Inflection: $f''(x) = 0$

10E&F) Maximum and Minimum Values

• Finding Max/Min: Set $f'(x) = 0$ and solve for x, then check endpoints and stationary points.

10G) Families of Functions

10H) Newton’s Method

• $x<em>{n+1} = x</em>n - \frac{f(x<em>n)}{f'(x</em>n)}$

Integration:

11A) The Area Under a Graph

11B&C) Anti-Differentiation: Indefinite Integrals

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

• $\int k \cdot f(x) dx = k \int f(x) dx$

• $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

11D) Integrating Exponential Functions

• $\int e^x dx = e^x + C$

• $\int e^{kx} dx = \frac{1}{k}e^{kx} + C$

11E) Definite Integrals

• $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$

11F) Finding the Area Under a Curve

• Area = $\int_a^b f(x) dx$

11G) Antiderivatives of Circular Functions

• $\int sin(x) dx = -cos(x) + C$

• $\int cos(x) dx = sin(x) + C$

11H) Integration by Recognition

11I) Area Between Curves

• Area = $\int_a^b [f(x) - g(x)] dx$, where f(x) > g(x) on $[a, b]$

11J) Applications of Integration

9B&C) Rules for Differentiation

• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$

  • e.g., If $f(x) = x^3$, then $f'(x) = 3x^2$

• Constant Multiple Rule: If $f(x) = c \cdot g(x)$, then $f'(x) = c \cdot g'(x)$

  • e.g., If $f(x) = 5x^2$, then $f'(x) = 5 \cdot 2x = 10x$

• Sum Rule: If $f(x) = u(x) + v(x)$, then $f'(x) = u'(x) + v'(x)$

  • e.g., If $f(x) = x^3 + sin(x)$, then $f'(x) = 3x^2 + cos(x)$

• Difference Rule: If $f(x) = u(x) - v(x)$, then $f'(x) = u'(x) - v'(x)$

  • e.g., If $f(x) = x^2 - cos(x)$, then $f'(x) = 2x - (-sin(x)) = 2x + sin(x)$

9E & 9F) The Chain Rule

• If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

  • e.g., If $y = sin(u)$ and $u = x^2$, then $\frac{dy}{dx} = cos(u) \cdot 2x = cos(x^2) \cdot 2x$

9G) Derivative of $e^x$

• If $f(x) = e^x$, then $f'(x) = e^x$

  • e.g., If $f(x) = e^x$, then $f'(x) = e^x$

• If $f(x) = e^{kx}$, then $f'(x) = ke^{kx}$

  • e.g., If $f(x) = e^{3x}$, then $f'(x) = 3e^{3x}$

9H) Derivative of $log_e(x)$

• If $f(x) = log_e(x) = ln(x)$, then $f'(x) = \frac{1}{x}$

  • e.g., If $f(x) = ln(x)$, then $f'(x) = \frac{1}{x}$

• If $f(x) = log_e(kx)$, then $f'(x) = \frac{1}{x}$

  • e.g., If $f(x) = ln(5x)$, then $f'(x) = \frac{1}{x}$

9I) Derivative of Trigonometric Functions

• If $f(x) = sin(x)$, then $f'(x) = cos(x)$

  • e.g., If $f(x) = sin(x)$, then $f'(x) = cos(x)$

• If $f(x) = cos(x)$, then $f'(x) = -sin(x)$

  • e.g., If $f(x) = cos(x)$, then $f'(x) = -sin(x)$

• If $f(x) = tan(x)$, then $f'(x) = sec^2(x) = 1 + tan^2(x)$

  • e.g., If $f(x) = tan(x)$, then $f'(x) = sec^2(x)$

9J) The Product Rule

• If $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$

  • e.g., If $f(x) = x^2 \cdot sin(x)$, then $f'(x) = 2x \cdot sin(x) + x^2 \cdot cos(x)$

9K) The Quotient Rule

• If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

  • e.g., If $f(x) = \frac{x^2}{cos(x)}$, then $f'(x) = \frac{2x \cdot cos(x) - x^2 \cdot (-sin(x))}{[cos(x)]^2}$

10A) Equations of Tangents and Normals

• Tangent: $y - y<em>1 = m(x - x</em>1)$, where m is the derivative at $(x<em>1, y</em>1)$

  • e.g., Find the tangent to $f(x) = x^2$ at $(2, 4)$. $f'(x) = 2x$, so $m = f'(2) = 4$. Tangent: $y - 4 = 4(x - 2)$

• Normal: $y - y<em>1 = -\frac{1}{m}(x - x</em>1)$, where -1/m is the negative reciprocal of the derivative at $(x<em>1, y</em>1)$

  • e.g., Find the normal to $f(x) = x^2$ at $(2, 4)$. $f'(x) = 2x$, so $m = f'(2) = 4$. Normal: $y - 4 = -\frac{1}{4}(x - 2)$

10B) Rates of Change

• Related Rates: Use chain rule to relate rates of change of different variables.

  • e.g., If the radius of a circle is increasing at a rate of 3 cm/s, find the rate at which the area is increasing when the radius is 6 cm. $A = \pi r^2$, $\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi (6)(3) = 36\pi$

10C&D) Stationary Points and Their Nature

• Stationary Points: Points where $f'(x) = 0$

  • e.g., Find the stationary points of $f(x) = x^3 - 3x$. $f'(x) = 3x^2 - 3 = 0$, so $x = \pm 1$

• Nature of Stationary Points:

  • Maximum: f''(x) < 0

    • e.g., For $f(x) = x^3 - 3x$, $f''(x) = 6x$. At $x = -1$, f''(-1) = -6 < 0, so $x = -1$ is a maximum.

  • Minimum: f''(x) > 0

    • e.g., For $f(x) = x^3 - 3x$, $f''(x) = 6x$. At $x = 1$, f''(1) = 6 > 0, so $x = 1$ is a minimum.

  • Point of Inflection: $f''(x) = 0$

    • e.g., For $f(x) = x^3$, $f''(x) = 6x$. At $x = 0$, $f''(0) = 0$, so $x = 0$ is a point of inflection.

10E&F) Maximum and Minimum Values

• Finding Max/Min: Set $f'(x) = 0$ and solve for x, then check endpoints and stationary points.

  • e.g., Find the maximum and minimum values of $f(x) = x^3 - 6x^2 + 5$ on the interval $[-1, 5]$. $f'(x) = 3x^2 - 12x = 0$, so $x = 0, 4$. Checking endpoints and stationary points: $f(-1) = -2$, $f(0) = 5$, $f(4) = -27$, $f(5) = -20$. Maximum value is 5, minimum value is -27.

10H) Newton’s Method

• $x<em>{n+1} = x</em>n - \frac{f(x<em>n)}{f'(x</em>n)}$

  • e.g., Approximate the root of $f(x) = x^3 - 2x - 5$ using Newton’s method with initial guess $x<em>0 = 2$. $f'(x) = 3x^2 - 2$. $x</em>1 = 2 - \frac{f(2)}{f'(2)} = 2 - \frac{-1}{10} = 2.1$.

11B&C) Anti-Differentiation: Indefinite Integrals

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

  • e.g., $\int x^2 dx = \frac{x^3}{3} + C$

• $\int k \cdot f(x) dx = k \int f(x) dx$

  • e.g., $\int 3x^2 dx = 3 \int x^2 dx = 3 \cdot \frac{x^3}{3} + C = x^3 + C$

• $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

  • e.g., $\int [x^2 + cos(x)] dx = \int x^2 dx + \int cos(x) dx = \frac{x^3}{3} + sin(x) + C$

11D) Integrating Exponential Functions

• $\int e^x dx = e^x + C$

  • e.g., $\int e^x dx = e^x + C$

• $\int e^{kx} dx = \frac{1}{k}e^{kx} + C$

  • e.g., $\int e^{2x} dx = \frac{1}{2}e^{2x} + C$

11E) Definite Integrals

• $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$

  • e.g., $\int<em>0^1 x^2 dx = \frac{x^3}{3} |</em>0^1 = \frac{1}{3} - 0 = \frac{1}{3}$

11F) Finding the Area Under a Curve

• Area = $\int_a^b f(x) dx$

  • e.g., Find the area under the curve $f(x) = x^2$ from 0 to 1. Area = $\int_0^1 x^2 dx = \frac{1}{3}$

11G) Antiderivatives of Circular Functions

• $\int sin(x) dx = -cos(x) + C$

  • e.g., $\int sin(x) dx = -cos(x) + C$

• $\int cos(x) dx = sin(x) + C$

  • e.g., $\int cos(x) dx = sin(x) + C$

11I) Area Between Curves

• Area = $\int_a^b [f(x) - g(x)] dx$, where f(x) > g(x) on $[a, b]$

  • e.g., Find the area between $f(x) = x^2$ and $g(x) = x$

Differentiation

Question 1:

Let $f(x) = x^3 \cdot sin(2x)$. Find $f'(x)$.

Solution:

Using the product rule, $f'(x) = u'v + uv'$

Let $u = x^3$ and $v = sin(2x)$

Then $u' = 3x^2$ and $v' = 2cos(2x)$

So, $f'(x) = 3x^2sin(2x) + x^3(2cos(2x)) = 3x^2sin(2x) + 2x^3cos(2x)$

Question 2:

Find the equation of the tangent to the curve $y = e^{2x}$ at the point where $x = 0$.

Solution:

When $x = 0$, $y = e^{2(0)} = e^0 = 1$. So the point is $(0, 1)$.

$\frac{dy}{dx} = 2e^{2x}$

At $x = 0$, the gradient $m = 2e^{2(0)} = 2e^0 = 2$

The equation of the tangent is $y - y<em>1 = m(x - x</em>1)$

So, $y - 1 = 2(x - 0)$

$y = 2x + 1$

Integration

Question 3:

Evaluate $\int_0^{\frac{\pi}{4}} cos(2x) dx$

Solution:

$\int cos(2x) dx = \frac{1}{2}sin(2x) + C$

So, $\int_0^{\frac{\pi}{4}} cos(2x) dx = \frac{1}{2} \Big[ sin(2 \cdot \frac{\pi}{4}) - sin(2 \cdot 0) \Big] = \frac{1}{2} \Big[ sin(\frac{\pi}{2}) - sin(0) \Big] = \frac{1}{2} \Big[ 1 - 0 \Big] = \frac{1}{2}$

Question 4:

Find the area between the curve $y = x^2$ and the line $y = 4x$ between $x = 0$ and $x = 4$.

Solution:

First, find the points of intersection:

$x^2 = 4x$

$x^2 - 4x = 0$

$x(x - 4) = 0$

So, $x = 0$ or $x = 4$

The area is given by $\int_0^4 (4x - x^2) dx$

$\int (4x - x^2) dx = 2x^2 - \frac{x^3}{3} + C$

\int_0^4 (4x - x^2) dx = \Big[ 2(4)^2 - \frac{(4

Differentiation

Question 1:

Let f(x) = x^3 \cdot sin(2x)$. Find$f'(x)$.</p><p>Solution:</p><p>Using the product rule,$f'(x) = u'v + uv'$</p><p>Let$u = x^3$and$v = sin(2x)$</p><p>Then$u' = 3x^2$and$v' = 2cos(2x)$</p><p>So,$f'(x) = 3x^2sin(2x) + x^3(2cos(2x)) = 3x^2sin(2x) + 2x^3cos(2x)$</p><p>Question 2:</p><p>Find the equation of the tangent to the curve$y = e^{2x}$at the point where$x = 0$.</p><p>Solution:</p><p>When$x = 0$,$y = e^{2(0)} = e^0 = 1$. So the point is$(0, 1)$.</p><p>$\frac{dy}{dx} = 2e^{2x}$</p><p>At$x = 0$, the gradient$m = 2e^{2(0)} = 2e^0 = 2$</p><p>The equation of the tangent is$y - y1 = m(x - x1)$</p><p>So,$y - 1 = 2(x - 0)$</p><p>$y = 2x + 1$</p><p>Integration</p><p>Question 3:</p><p>Evaluate$\int_0^{\frac{\pi}{4}} cos(2x) dx$</p><p>Solution:</p><p>$\int cos(2x) dx = \frac{1}{2}sin(2x) + C$</p><p>So,$\int_0^{\frac{\pi}{4}} cos(2x) dx = \frac{1}{2} \Big[ sin(2 \cdot \frac{\pi}{4}) - sin(2 \cdot 0) \Big] = \frac{1}{2} \Big[ sin(\frac{\pi}{2}) - sin(0) \Big] = \frac{1}{2} \Big[ 1 - 0 \Big] = \frac{1}{2}$</p><p>Question 4:</p><p>Find the area between the curve$y = x^2$and the line$y = 4x$between$x = 0$and$x = 4$.</p><p>Solution:</p><p>First, find the points of intersection:</p><p>$x^2 = 4x$</p><p>$x^2 - 4x = 0$</p><p>$x(x - 4) = 0$</p><p>So,$x = 0$or$x = 4$</p><p>The area is given by$\int_0^4 (4x - x^2) dx$</p><p>$\int (4x - x^2) dx = 2x^2 - \frac{x^3}{3} + C$</p><p>$\int_0^4 (4x - x^2) dx = \Big[ 2(4)^2 - \frac{(4