Glossary

5.3 Rates of change and motion

2 min read•july 22, 2024

Derivatives are the key to understanding how things change. They help us calculate rates of change, like and , which are crucial in physics and engineering. We use them to analyze motion and predict how objects move.

are the tools we use to find derivatives. From the basic to the more complex , these techniques let us tackle a wide range of functions and solve real-world problems involving motion and change.

Rates of Change and Motion

Derivative as rate of change

Top images from around the web for Derivative as rate of change

  • Acceleration - Wikipedia, the free encyclopedia View original

    Is this image relevant?

  • Average and Instantaneous Acceleration – University Physics Volume 1 View original

    Is this image relevant?

  • Acceleration - Wikipedia, the free encyclopedia View original

    Is this image relevant?

1 of 3

Top images from around the web for Derivative as rate of change

  • Acceleration - Wikipedia, the free encyclopedia View original

    Is this image relevant?

  • Average and Instantaneous Acceleration – University Physics Volume 1 View original

    Is this image relevant?

  • Acceleration - Wikipedia, the free encyclopedia View original

    Is this image relevant?

1 of 3
  • Represents measures how quickly a function changes at a specific point
  • Velocity rate of change of position with respect to time of position function gives velocity (distance traveled over time)
  • Acceleration rate of change of velocity with respect to time derivative of velocity function gives acceleration (change in speed over time)

Applications of differentiation rules

  • Power rule find derivatives of polynomial functions ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1} (square function x2x^2, cube function x3x^3)
  • differentiate functions with coefficients ddx(cf(x))=cddxf(x)\frac{d}{dx}(cf(x)) = c\frac{d}{dx}f(x), cc is a constant (2x, 3x)
  • differentiate combinations of functions ddx(f(x)±g(x))=ddxf(x)±ddxg(x)\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x) (polynomials, trigonometric functions)
  • more complex functions ddx(f(x)g(x))=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x) (polynomial multiplied by exponential function)
  • more complex functions ddx(f(x)g(x))=g(x)ddxf(x)−f(x)ddxg(x)[g(x)]2\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^2} (rational functions)
  • Chain rule composite functions ddxf(g(x))=f′(g(x))g′(x)\frac{d}{dx}f(g(x)) = f'(g(x))g'(x) (function inside another function like sin⁡(x2)\sin(x^2))

Position, velocity, and acceleration relationships

  • Position location of an object at a given time (distance from origin)
  • Velocity first derivative of position with respect to time represents rate of change of position indicates speed and direction of motion
  • Acceleration second derivative of position or first derivative of velocity with respect to time represents rate of change of velocity indicates how quickly velocity changes

Instantaneous motion from position functions

  • Given position function s(t)s(t), find velocity by taking first derivative velocity function: v(t)=ddts(t)v(t) = \frac{d}{dt}s(t)
  • Find acceleration by taking second derivative of position or first derivative of velocity acceleration function: a(t)=d2dt2s(t)=ddtv(t)a(t) = \frac{d^2}{dt^2}s(t) = \frac{d}{dt}v(t)
  • Evaluate velocity and acceleration functions at specific time to determine instantaneous values (velocity at t=2 seconds, acceleration at t=3 seconds)

Motion analysis with derivatives

  1. Identify given information and desired quantity to solve for (initial position, velocity, time)
  2. Set up position function based on problem description (quadratic for constant acceleration, trigonometric for periodic motion)
  3. Use differentiation rules to find velocity and acceleration functions
  4. Evaluate functions at specific times or solve for unknown variables (position at t=5 seconds, time when velocity is zero)
  5. Interpret results in context of problem
    • Positive velocity indicates motion in positive direction, negative velocity indicates motion in negative direction
    • Positive acceleration represents increasing velocity, negative acceleration represents decreasing velocity

Key Terms to Review (21)

Acceleration: Acceleration is the rate of change of velocity of an object with respect to time. It reflects how quickly an object is speeding up, slowing down, or changing direction. This concept is crucial as it connects to the behavior of moving objects and helps in understanding their dynamics, which involves analyzing slopes, rates of change, and even the implications of higher-order derivatives.
Average rate of change: The average rate of change of a function over an interval measures how much the function's output value changes relative to the input value changes across that interval. This concept is crucial in understanding how functions behave between two points and plays a significant role in motion analysis and the application of the Mean Value Theorem, which bridges the gap between average and instantaneous rates of change.
Chain Rule: The chain rule is a fundamental technique in calculus used to differentiate composite functions, allowing us to find the derivative of a function that is made up of other functions. This rule is crucial for understanding how different rates of change are interconnected and enables us to tackle complex differentiation problems involving multiple layers of functions.
Constant Multiple Rule: The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of that function. This rule is fundamental in calculus as it allows for simplification when taking derivatives, making it easier to analyze and understand functions across various contexts.
Derivative: A derivative represents the rate at which a function changes at any given point, essentially capturing the slope of the tangent line to the curve of that function. This concept is fundamental in understanding how functions behave, especially when analyzing instantaneous rates of change, optimizing functions, and solving real-world problems involving motion and growth.
Differentiation Rules: Differentiation rules are established guidelines in calculus that provide systematic methods for calculating the derivative of a function. These rules streamline the process of finding rates of change, which is essential for understanding motion and other dynamic processes. By applying these rules, one can efficiently determine how a function behaves locally, revealing important information about its slope, concavity, and overall behavior.
Dy/dx: The notation $$\frac{dy}{dx}$$ represents the derivative of a function, indicating the rate at which the dependent variable $$y$$ changes with respect to the independent variable $$x$$. This concept is essential for understanding how functions behave and helps in solving problems related to tangents, slopes, and rates of change. The derivative encapsulates the instantaneous rate of change, allowing for the analysis of motion and the dynamics of systems.
F'(x): The notation f'(x) represents the derivative of the function f at the point x, indicating the rate at which the function's value changes as x changes. This concept is crucial for understanding how functions behave, particularly in determining slopes of tangent lines, rates of change, and overall function behavior, which are foundational in various applications such as motion analysis and optimization problems.
Instantaneous Rate of Change: The instantaneous rate of change refers to the rate at which a function is changing at any specific point, which can be understood as the slope of the tangent line to the graph of the function at that point. This concept is essential for understanding how functions behave at particular values and is closely related to the derivative, which formalizes this idea mathematically. In practical terms, it helps in analyzing motion, understanding changes in variables, and applying important theorems related to functions.
Limit definition of a derivative: The limit definition of a derivative is a fundamental concept in calculus that describes the instantaneous rate of change of a function at a particular point. It is formally expressed as the limit of the average rate of change of the function as the interval approaches zero. This definition connects to various applications, such as understanding how functions behave in motion and determining their rates of change over time.
Linear functions: Linear functions are mathematical expressions that describe a straight line when graphed on a coordinate plane, represented in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. This concept is crucial when analyzing how quantities change in relation to each other, making it essential for understanding motion and rates of change, as well as tackling related rates problems.
Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change of the function over that interval. This theorem provides a bridge between the behavior of a function and its derivatives, showing how slopes relate to overall changes.
Optimization problems: Optimization problems involve finding the best solution or outcome from a set of possible choices, typically subject to certain constraints. These problems often require determining maximum or minimum values of a function, and are closely linked to concepts like rates of change, the application of derivatives, and methods for evaluating functions over defined intervals.
Power Rule: The power rule is a fundamental principle in calculus that provides a quick way to differentiate functions of the form $$f(x) = x^n$$, where $$n$$ is any real number. This rule states that the derivative of such a function is given by multiplying the exponent by the coefficient and then reducing the exponent by one, resulting in $$f'(x) = n imes x^{n-1}$$. It connects to concepts of slopes and rates of change, making it essential for understanding how functions behave.
Product Rule: The product rule is a fundamental principle in calculus that provides a method for finding the derivative of the product of two functions. It states that if you have two functions, say $$u(x)$$ and $$v(x)$$, the derivative of their product can be calculated using the formula: $$ (uv)' = u'v + uv' $$, where $$u'$$ and $$v'$$ are the derivatives of $$u$$ and $$v$$ respectively. This concept is crucial in understanding how derivatives work when dealing with more complex functions that are products of simpler ones.
Quadratic functions: Quadratic functions are polynomial functions of degree two, represented by the standard form $$f(x) = ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. These functions create a parabolic graph that can open upwards or downwards, depending on the sign of 'a'. Quadratic functions are essential in various mathematical applications including motion analysis, optimization problems, and solving related rates challenges.
Quotient Rule: The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. It states that if you have a function that can be expressed as $$f(x) = \frac{g(x)}{h(x)}$$, where both $$g$$ and $$h$$ are differentiable, then the derivative is given by $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$. This rule connects to understanding how rates of change behave in division scenarios, as well as its application alongside other rules such as the product rule and logarithmic differentiation.
Related Rates Problems: Related rates problems involve finding the rate at which one quantity changes with respect to another, often using the chain rule in calculus. These problems typically relate two or more variables that are dependent on each other, allowing us to derive one rate of change from another. They are often set in real-world contexts, such as physics and engineering, making them essential for understanding motion and rates of change in practical scenarios.
Rolle's Theorem: Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval between the endpoints, and if the function takes the same value at both endpoints, then there exists at least one point in the open interval where the derivative of the function is zero. This theorem highlights a crucial relationship between differentiability, continuity, and the behavior of functions on intervals.
Sum and Difference Rules: The sum and difference rules are fundamental concepts in calculus that provide a way to find the derivatives of sums and differences of functions. Specifically, these rules state that the derivative of the sum of two functions is the sum of their derivatives, and the derivative of the difference of two functions is the difference of their derivatives. Understanding these rules is essential for analyzing rates of change, particularly when dealing with motion problems, where multiple variables often combine.
Velocity: Velocity is a vector quantity that refers to the rate of change of an object's position with respect to time, indicating both speed and direction. Understanding velocity is crucial because it helps analyze motion more comprehensively than speed alone, allowing for insights into how fast something is moving and in which direction. This concept is foundational in calculus, particularly when exploring the slope of tangent lines and how rates of change relate to motion over time.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide