Calculus

Calculus Lecture for Ninth Grade

Unit 1: Introduction to Functions

1. What is a Function?

   • A function is a relationship between two sets of numbers where each input (x-value) has exactly one output (y-value).

   • We write it as $y = f(x)$$$y = f(x)$$, where $x$$$x$$ is the input and $f(x)$$$f(x)$$ is the output.

   • Example: $f(x) = 2x + 3$$$f(x) = 2x + 3$$

     • If $x = 1$$$x = 1$$, then $f(1) = 2(1) + 3 = 5$$$f(1) = 2(1) + 3 = 5$$. So when the input is 1, the output is 5.

2. Types of Functions

   • Linear Functions: Have the form $f(x) = mx + b$$$f(x) = mx + b$$, where $m$$$m$$ is the slope and $b$$$b$$ is the y-intercept.

     • Example: $f(x) = 3x + 2$$$f(x) = 3x + 2$$

   • Quadratic Functions: Have the form $f(x) = ax^2 + bx + c$$$f(x) = ax^2 + bx + c$$, where $a$$$a$$, $b$$$b$$, and $c$$$c$$ are constants.

     • Example: $f(x) = x^2 - 4x + 1$$$f(x) = x^2 - 4x + 1$$

   • Polynomial Functions: Include linear and quadratic functions but can have higher powers of $x$$$x$$.

     • Example: $f(x) = x^3 + 2x^2 - x + 5$$$f(x) = x^3 + 2x^2 - x + 5$$

3. Graphing Functions

   • To graph a function, we plot points $(x, f(x))$$$(x, f(x))$$ on a coordinate plane.

   • For a linear function, you only need two points to draw the entire line.

   • For other functions, plot enough points to see the shape of the curve.

   • Example: Graph $f(x) = x + 1$$$f(x) = x + 1$$

     • When $x = 0$$$x = 0$$, $f(0) = 1$$$f(0) = 1$$. Plot the point $(0, 1)$$$(0, 1)$$.

     • When $x = 1$$$x = 1$$, $f(1) = 2$$$f(1) = 2$$. Plot the point $(1, 2)$$$(1, 2)$$.

     • Draw a line through these points.

Unit 2: Introduction to Limits

1. What is a Limit?

   • A limit describes the value that a function approaches as the input gets closer and closer to some value.

   • Notation: $\lim_{x \to a} f(x) = L$$$\lim_{x \to a} f(x) = L$$ means as $x$$$x$$ gets close to $a$$$a$$, $f(x)$$$f(x)$$ gets close to $L$$$L$$.

   • Example: $\lim_{x \to 2} (x + 3) = 5$$$\lim_{x \to 2} (x + 3) = 5$$

     • As $x$$$x$$ gets closer to 2, $x + 3$$$x + 3$$ gets closer to 5.

2. Evaluating Limits

   • Direct Substitution: If the function is continuous at the point, just plug in the value.

     • Example: $\lim_{x \to 1} (x^2 + 2x + 1) = (1)^2 + 2(1) + 1 = 4$$$\lim_{x \to 1} (x^2 + 2x + 1) = (1)^2 + 2(1) + 1 = 4$$

   • Factoring: If direct substitution gives an indeterminate form like $\frac{0}{0}$$$\frac{0}{0}$$, try factoring.

     • Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$

       • $= \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}$$$= \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}$$

       • $= \lim_{x \to 2} (x + 2) = 4$$$= \lim_{x \to 2} (x + 2) = 4$$

3. One-Sided Limits

   • The limit from the left: $\lim_{x \to a^-} f(x)$$$\lim_{x \to a^-} f(x)$$

   • The limit from the right: $\lim_{x \to a^+} f(x)$$$\lim_{x \to a^+} f(x)$$

   • For a limit to exist, both one-sided limits must exist and be equal.

Unit 3: Introduction to Derivatives

1. What is a Derivative?

   • The derivative of a function gives the slope of the tangent line at any point on the function.

   • It measures the instantaneous rate of change of a function.

   • Notation: If $y = f(x)$$$y = f(x)$$, the derivative is written as $f'(x)$$$f'(x)$$ or $\frac{dy}{dx}$$$\frac{dy}{dx}$$.

2. The Definition of the Derivative

   • $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

   • This formula calculates the slope of the tangent line.

3. Basic Differentiation Rules

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

     • Example: If $f(x) = x^3$$$f(x) = x^3$$, then $f'(x) = 3x^2$$$f'(x) = 3x^2$$.

   • Constant Multiple Rule: If $f(x) = cf(x)$$$f(x) = cf(x)$$, then $f'(x) = cf'(x)$$$f'(x) = cf'(x)$$, where $c$$$c$$ is a constant.

     • Example: If $f(x) = 5x^2$$$f(x) = 5x^2$$, then $f'(x) = 5(2x) = 10x$$$f'(x) = 5(2x) = 10x$$.

   • Sum/Difference Rule: If $f(x) = u(x) \pm v(x)$$$f(x) = u(x) \pm v(x)$$, then $f'(x) = u'(x) \pm v'(x)$$$f'(x) = u'(x) \pm v'(x)$$.

     • Example: If $f(x) = x^3 + 2x$$$f(x) = x^3 + 2x$$, then $f'(x) = 3x^2 + 2$$$f'(x) = 3x^2 + 2$$.

Unit 4: Applications of Derivatives

1. Finding Slope of Tangent Lines

   • To find the slope of the tangent line at a point $x = a$$$x = a$$, calculate $f'(a)$$$f'(a)$$.

   • Example: Find the slope of the tangent line to $f(x) = x^2$$$f(x) = x^2$$ at $x = 3$$$x = 3$$.

     • $f'(x) = 2x$$$f'(x) = 2x$$, so $f'(3) = 2(3) = 6$$$f'(3) = 2(3) = 6$$.

2. Increasing and Decreasing Functions

   • If $f'(x) > 0$$$f'(x) > 0$$ on an interval, then $f(x)$$$f(x)$$ is increasing on that interval.

   • If $f'(x) < 0$$$f'(x) < 0$$ on an interval, then $f(x)$$$f(x)$$ is decreasing on that interval.

   • If $f'(x) = 0$$$f'(x) = 0$$ at a point, then that point may be a local maximum or minimum.

3. Maximum and Minimum Values

   • To find local maxima and minima, find where $f'(x) = 0$$$f'(x) = 0$$ or where $f'(x)$$$f'(x)$$ is undefined.

   • Test these points to determine if they are maxima, minima, or neither.

   • Example: Find the local maxima and minima of $f(x) = x^3 - 6x^2 + 5$$$f(x) = x^3 - 6x^2 + 5$$.

     • $f'(x) = 3x^2 - 12x = 3x(x - 4)$$$f'(x) = 3x^2 - 12x = 3x(x - 4)$$

     • $f'(x) = 0$$$f'(x) = 0$$ when $x = 0$$$x = 0$$ and $x = 4$$$x = 4$$.

Unit 5: Introduction to Integrals

1. What is an Integral?

   • An integral is the reverse process of differentiation.

   • It finds the area under a curve.

   • Notation: $\int f(x) dx = F(x) + C$$$\int f(x) dx = F(x) + C$$, where $F(x)$$$F(x)$$ is the antiderivative of $f(x)$$$f(x)$$ and $C$$$C$$ is the constant of integration.

2. Basic Integration Rules

   • Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, where $n \neq -1$$$n \neq -1$$.

     • Example: $\int x^2 dx = \frac{x^3}{3} + C$$$\int x^2 dx = \frac{x^3}{3} + C$$.

   • Constant Multiple Rule: $\int cf(x) dx = c \int f(x) dx$$$\int cf(x) dx = c \int f(x) dx$$, where $c$$$c$$ is a constant.

     • Example: $\int 5x^3 dx = 5 \int x^3 dx = 5(\frac{x^4}{4}) + C = \frac{5x^4}{4} + C$$$\int 5x^3 dx = 5 \int x^3 dx = 5(\frac{x^4}{4}) + C = \frac{5x^4}{4} + C$$.

   • Sum/Difference Rule: $\int [u(x) \pm v(x)] dx = \int u(x) dx \pm \int v(x) dx$$$\int [u(x) \pm v(x)] dx = \int u(x) dx \pm \int v(x) dx$$.

     • Example: $\int (x^2 + x) dx = \int x^2 dx + \int x dx = \frac{x^3}{3} + \frac{x^2}{2} + C$$$\int (x^2 + x) dx = \int x^2 dx + \int x dx = \frac{x^3}{3} + \frac{x^2}{2} + C$$.

3. Definite Integrals

   • Definite integrals have limits of integration:

   • $$\int_{a}^{b} f(x) dx = F(b) -