How to find the derivative of $f(x) = x^2$ using the limit definition?
1. Plug into the limit definition: $\lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$. 2. Expand: $\lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$. 3. Simplify: $\lim_{h \to 0} \frac{2xh + h^2}{h}$. 4. Factor and cancel: $\lim_{h \to 0} (2x + h)$. 5. Evaluate the limit: $2x$.
How to find the tangent line to $f(x) = x^2$ at $x = 1$?
1. Find $f'(x)$: $f'(x) = 2x$. 2. Find the slope at $x = 1$: $f'(1) = 2$. 3. Find the y-coordinate at $x = 1$: $f(1) = 1$. 4. Use point-slope form: $y - 1 = 2(x - 1)$.
How to find where the tangent line is horizontal for $f(x) = x^3 - 3x$?
1. Find $f'(x)$: $f'(x) = 3x^2 - 3$. 2. Set $f'(x) = 0$: $3x^2 - 3 = 0$. 3. Solve for $x$: $x = \pm 1$.
How to evaluate $\lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}$?
1. Find a common denominator: $\lim_{h \to 0} \frac{\frac{x - (x+h)}{x(x+h)}}{h}$. 2. Simplify: $\lim_{h \to 0} \frac{\frac{-h}{x(x+h)}}{h}$. 3. Multiply by the reciprocal: $\lim_{h \to 0} \frac{-h}{x(x+h)h}$. 4. Cancel $h$: $\lim_{h \to 0} \frac{-1}{x(x+h)}$. 5. Evaluate the limit: $\frac{-1}{x^2}$.
How to find the derivative of $y = 3x^2 + 4x$ using the limit definition?
1. Plug into the limit definition: $\lim_{h \to 0} \frac{[3(x+h)^2 + 4(x+h)] - [3x^2 + 4x]}{h}$. 2. Expand: $\lim_{h \to 0} \frac{3x^2 + 6xh + 3h^2 + 4x + 4h - 3x^2 - 4x}{h}$. 3. Simplify: $\lim_{h \to 0} \frac{6xh + 3h^2 + 4h}{h}$. 4. Factor and cancel: $\lim_{h \to 0} (6x + 3h + 4)$. 5. Evaluate the limit: $6x + 4$.
Define the derivative of a function.
The instantaneous rate of change of a function at a given point.
What does $f'(x)$ represent?
The derivative of the function $f(x)$, representing the slope of the tangent line at $x$.
Define instantaneous rate of change.
The rate of change of a function at a specific instant in time, equivalent to the derivative at that point.
What is a tangent line?
A line that touches a curve at a single point and has the same slope as the curve at that point.
What is the limit definition of the derivative?
A formal way of defining the derivative as the limit of the difference quotient as h approaches zero.
Define $y'$ notation.
A notation representing the first derivative of $y$ with respect to the independent variable (usually $x$).
Define $\frac{dy}{dx}$ notation.
Leibniz's notation for the derivative of $y$ with respect to $x$, representing an infinitesimally small change in $y$ divided by an infinitesimally small change in $x$.
What does 'rate of change' mean in calculus?
How one quantity changes in relation to another quantity, often represented by the derivative.
What is the difference quotient?
The expression $\frac{f(x + h) - f(x)}{h}$, used to calculate the average rate of change of a function over an interval of length $h$.
What is the meaning of slope of a curve?
The slope of the tangent line to the curve at a specific point, representing the instantaneous rate of change at that point.
Explain the relationship between a derivative and a tangent line.
The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point.
What does the derivative tell us about the behavior of a function?
The derivative indicates whether the function is increasing or decreasing, and the rate at which it is changing.
Explain the concept of a limit in the context of derivatives.
The limit allows us to find the instantaneous rate of change by considering what happens as the interval over which we calculate the average rate of change becomes infinitesimally small.
Why is the limit definition of the derivative important?
It provides the fundamental basis for understanding and calculating derivatives, and it connects the concept of slope to curves.
What is the significance of $h \to 0$ in the limit definition?
It means we are considering the slope of the secant line as the two points get infinitely close together, effectively becoming the tangent line.
Explain the concept of differentiability.
A function is differentiable at a point if its derivative exists at that point. This implies the function is continuous and has a well-defined tangent line.
What is the relationship between differentiability and continuity?
If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.
Explain how the derivative relates to real-world rates of change.
The derivative can model various real-world phenomena, such as velocity (rate of change of position), acceleration (rate of change of velocity), and rates of chemical reactions.
What does it mean for a tangent line to be horizontal?
A horizontal tangent line indicates that the derivative is zero at that point, implying a local maximum, local minimum, or a stationary point.
How do different notations of derivatives relate to each other?
$y'$, $f'(x)$, and $\frac{dy}{dx}$ all represent the same concept: the derivative of a function. They are used interchangeably depending on the context.
