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Drill: Find dy / dx

Drill: Find dy / dx. y = sin(x 2) y = sec 2 x – tan 2 x y = 2 x y’ = 2cos(x 2 ) Let u = sec x and v = tan x du = secxtanx and dv = sec 2 x y = u 2 – v 2. y’ = 2u du - 2v dv y’ = 2(sec x)( secxtanx ) - 2( tanx )(sec 2 x) = 0 3. y’ =2 x ln 2.

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Drill: Find dy / dx

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  1. Drill: Find dy/dx • y = sin(x2) • y = sec2x – tan2 x • y = 2x • y’ = 2cos(x2 ) • Let u = sec x and v = tan x du = secxtanx and dv = sec2 x y = u2 – v2 y’ = 2u du - 2v dv y’ = 2(sec x)(secxtanx) - 2(tanx)(sec2x) = 0 3. y’ =2xln 2

  2. Fundamental Theorem of Calculus Lesson 5.4 Day 1 Homework: p. 302/3: 1-25 ODD

  3. Objectives • Students will be able to • apply the Fundamental Theorem of Calculus. • understand the relationship between the derivative and definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

  4. The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function has a derivative at every point x in [a, b], and

  5. Example 1 Applying the Fundamental Theorem Find Note: It does not matter what a is, as long as b is ‘x.’

  6. Example 2 The Fundamental Theorem with the Chain Rule Find

  7. Example 3 The Fundamental Theorem with the Chain Rule

  8. Example 4 The Fundamental Theorem with the Chain Rule Find

  9. Example 5 Variable Lower Limits of Integration • Find dy/dx • Need to use the rules of integrals. Remember: • So, • dy/dx = -3xsin(x) • Find dy/dx

  10. Example 5 Constructing a Function with Given Derivative and Value • Find a function y = f(x) with derivative dy/dx = tanx that satisfies the condition f(3) = 5. • To construct a function with derivative of tanx (always let the lower bound be the given value of x.) • Remember that , • So, if x = 3, then • Therefore, we would only need to add 5 to construct a function whose derivative is tan(x) and where f(3) = 5

  11. Drill: Construct a function of the form that satisfies the given conditions.

  12. The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], and F is any antiderivative of f on the interval, then This part is also called the Integral Evaluation Theorem.

  13. Example 6 Evaluating an Integral Evaluate using an antiderivative.

  14. Example 6 Evaluating an Integral Evaluate using an antiderivative.

  15. How to Find Area Analytically To find the area between the graph of y = f(x) and the x-axis over the interval [a,b] analytically, • Graph the equation. • Partition [a,b] with the zeros of f. • Integrate f over each subinterval • Add the absolute values of the integrals

  16. Find the area of the region between the curve y = 4- x2, [0, 3], and the x-axis • Graph the equation. • Partition [a,b] with the zeros of f. • Integrate f over each subinterval • Add the absolute values of the integrals • The first region, where the graph is positive, is from [0, 2]. The next region, where the graph is negative, is from [2, 3]

  17. How to find Total Area on the Calculator • To find the area between the graph of y = f(x) and the x-axis over the interval [a, b] numerically, evaluate fnInt(|f(x)|, x, a, b) on the calculator. • Try the previous example: fnInt(|4-x2|, x, 0, 3) • 7.66666667

  18. Homework • Page 303: 27-47 (ODD)

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