AP Calculus- BC College Board

This subject is broken down into 109 topics in 10 modules:

  1. Analytical Applicatiions of Differentiation 12 topics
  2. Applications of Integration 13 topics
  3. Contextual Applications of Differentiation 7 topics
  4. Differential Equations 9 topics
  5. Differentiation: Composite, Implicit, and Inverse Functions 6 topics
  6. Differentiation: Definition and Basic Derivative Rules 10 topics
  7. Infinite Sequences and Series 15 topics
  8. Integration and Accumulation of Change 14 topics
  9. Limits and Continuity 15 topics
  10. Parametric Equations, Polar Coordinates, and Vector Valued Functions 8 topics
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This page was last modified on 28 September 2024.

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Calculus- BC

Analytical Applicatiions of Differentiation

Connecting to a Function, its first derivative, and its second derivative

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Connecting to a Function, its first derivative, and its second derivative

Understanding Differentiation

  • Differentiation provides a method to compute rates of change and determine the slope of a function at any point.
  • The first derivative of a function, denoted as f'(x) or df/dx, represents the instantaneous rate of change at a point, or the slope of the tangent line to the curve at that point.
  • A function will be increasing where its first derivative is positive, and decreasing where its first derivative is negative.
  • Local maxima and minima of a function can be found where its first derivative equals zero.

Connecting a Function and Its First Derivative

  • If f'(x) > 0, then the function f(x) is rising at x.
  • If f'(x) = 0, then the function f(x) has a horizontal tangent at x.
  • If f'(x) < 0, then the function f(x) is falling at x.
  • Points of inflection are points at which the function changes concavity (i.e., switches between curving up and curving down). You can locate these by finding where the first derivative changes sign.

Understanding Second Derivative

  • The second derivative, denoted as f''(x), provides information about the concavity or curvature of the graph.
  • Specifically, if the second derivative is positive at a point, the function is concave up at that point. Conversely, if the second derivative is negative, the function is concave down.
  • More intuitively, when a function is concave up, it curves upwards like the shape of a bowl. When a function is concave down, it curves downwards.

Connecting a Function, its First and Second Derivatives

  • A function itself gives us information about the shape, position, and values of particular points.
  • The first derivative gives us information about the rate of change of the function, identifying where the function has local extrema and inflection points.
  • The second derivative confers information about the concavity or curvature of the function. This helps further classify local extrema and also identify inflection points.

Course material for Calculus- BC, module Analytical Applicatiions of Differentiation, topic Connecting to a Function, its first derivative, and its second derivative

Calculus- BC

Differentiation: Definition and Basic Derivative Rules

The Product Rule

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The Product Rule

Understanding the Product Rule

  • The Product Rule is a basic derivative rule that is used when calculating the derivative of the product of two functions.
  • This rule says that the derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.
  • Mathematically, if f(x) = g(x)h(x), the derivative is f'(x) = g'(x)h(x) + g(x)h'(x).

Applying the Product Rule

  • When differentiating a product of two functions, first take the derivative of one function and leave the other function as it is. Then, do the reverse - leave the first function as it is and differentiate the second.
  • An easy way to remember this rule is to say that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second.
  • It's important to note that the order of the functions in the product does not affect the result due to the Commutative Property of Addition. g'(x)h(x) + g(x)h'(x) is the same as g(x)h'(x) + g'(x)h(x).

Practising the Product Rule

  • To get a good grasp of the Product Rule, it's crucial to practise differentiating various products of functions using this rule.
  • This practise not only helps you understand the usage of the rule better, but it also allows you to become faster and more accurate in your calculations during tests.
  • Always double check your work, particularly the calculation of derivatives. Mistakes can often occur in these steps and lead to wrong answers.

Useful Tips

  • The Product Rule is just one of several essential rules that you need to know for differentiation. Make sure you are also comfortable with the other fundamental rules, such as the Power Rule, Chain Rule, and Quotient Rule.
  • Understanding the concept behind the Product rule will help you in the long term to make sure not just to memorise the formula but understand why it works the way it does.
  • Try to identify problems that require the application of the Product Rule. If you see an equation where one function is being multiplied by another, it's a strong indication to use the rule.

Course material for Calculus- BC, module Differentiation: Definition and Basic Derivative Rules, topic The Product Rule

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