AP Calculus BC. Chapter 3: Derivatives 3.3: Rules for Differentiation

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1 AP Calculus BC Chapter 3: Derivatives 3.3: Rules for Differentiation

2 Essential Question & Why: Essential Question: How can I (easily) differentiate a polynomial or rational function? Why? Polynomial functions model many real-world phenomena and are commonly used to approximate other, more complicated, functions.

3 Learning Target: Use rules of differentiation to calculate derivatives, including second and higher-order derivatives.

4 Learning Objective: Big Idea 2: Derivatives Enduring Understanding 2.1: Students will understand that the derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies. Learning Objective 2.1C: Students will be able to calculate derivatives. Essential Knowledge 2.1C3: Students will know that sums, differences, products, and quotients of functions can be differentiated using derivative rules.

5 Learning Objective: Big Idea 2: Derivatives EU 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies. LO 2.1D: Determine higher order derivatives. EK 2.1D1: Students will know that differentiating f produces the second derivative f, provided the derivative of f exists; repeating this process higher order derivatives of f.

6 Learning Objective: Big Idea 2: Derivatives EU 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies. LO 2.1D: Determine higher order derivatives. EK 2.1D2: Students will know that higher order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include d 2 y f (x), and y. dx 2 Higher order derivatives can be denoted d n y dx n or f (n) (x).

7 Mathematical Practices for AP Calculus MPAC 3: Implementing algebraic/computational processes. Students can complete algebraic/computational processes correctly.

8 Quote for today: To count, tally, calculate, and measure; to compute, reckon, encode, decode, classify, and quantify; to enumerate, estimate, and tabulate; to arrange in a sequence or hierarchy or order: number is essential to our management and understanding of life. Denis Guedj ( ), in Numbers: The Universal Language

9 Rules for Differentiation: Derivative of a Constant Function: If f is the function with the constant value c, then df dx = d dx c ( ) = 0 Power Rule for Positive Integer Powers of x: If n is a positive integer, then d dx xn ( ) = nx n 1

10 Rules for Differentiation: The Constant Multiple Rule: If u is a differentiable function of x and c is a constant, then d ( dx cu ) = c du dx

11 Rules for Differentiation: The Sum and Difference Rule: If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, d ( dx u ± v ) = du dx ± dv dx

12 Rules for Differentiation: The Product Rule: The product of two differentiable functions u and v is differentiable and d ( dx uv ) = u dv dx + v du dx

13 Rules for Differentiation: The Quotient Rule: At a point where v 0, the quotient y = u / v of two differentiable functions is differentiable and d dx! u # $ " v % & = v du dx u dv dx v 2

14 Rules for Differentiation: Power Rule for Negative Integer Powers of x: If n is a negative integer and x 0, then d ( ) = nx n 1 dx xn

15 Assignments for 10/5: WU 3.3A: Q28 & Q29. CW 3.3: #3, 6, 9, 11, 12, 15, 16, 22, 28, & 30. HW 3.3B: Review 3.3 and complete the notes page.

AP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines

AP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines AP Calculus BC Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines Essential Questions & Why: Essential Questions: What is the difference between average and instantaneous rates of