Math 1 Lecture 20. Dartmouth College. Wednesday

Transcription

1 Math 1 Lecture 20 Dartmouth College Wenesay

2 Contents Reminers/Announcements Last Time Derivatives of Trigonometric Functions

3 Reminers/Announcements WebWork ue Friay x-hour problem session rop in Thursay Quiz Monay Turn in MiQuarter surveys Turn in Written Homework Office hours toay 1pm - 3pm

4 Last Time Derivatives of proucts an quotients Higher erivatives

5 Last Time Derivatives of proucts an quotients Higher erivatives Let s look at a couple practice problems from last time...

6 For what values of x oes the graph of f (x) = x 3 + 3x 2 + x + 3 have a horizontal tangent line?

7 For what values of x oes the graph of f (x) = x 3 + 3x 2 + x + 3 have a horizontal tangent line? Also, compute 2 x 2 (f (x)), 3 x 3 (f (x)), an 4 (f (x)). x 4

8 Suppose h(2) = 4, h (2) = 3, an h (2) = 1. Compute the following values: x 2 x 2 ( h(x) x ( h(x) 5 ) x=2 ) x=2

9 y π 2 π 3π 2π cos x x y π 2 π 3π 2π sin x x

10 y π 2 π 3π 2π cos x x y π 2 π 3π 2π sin x x Let f (x) = sin(x).

11 y π 2 π 3π 2π cos x x y π 2 π 3π 2π sin x x Let f (x) = sin(x). What is f (0)?

12 y π 2 π 3π 2π cos x x y π 2 π 3π 2π sin x x Let f (x) = sin(x). What is f (0)? What is f (π/2)?

13 y π 2 π 3π 2π cos x x y π 2 π 3π 2π sin x x Let f (x) = sin(x). What is f (0)? What is f (π/2)? How oes this value relate to the function cos(x)?

14 By observations on the previous slie we might guess that (sin(x)) = cos(x) x (cos(x)) = sin(x). x

15 By observations on the previous slie we might guess that (sin(x)) = cos(x) x (cos(x)) = sin(x). x It is beyon the scope of this course to verify these via the efinition, but we o nee to be able to use these rules to compute examples.

16 Note that we can compute the erivatives of all 6 trigonometric functions using the previous slie an the quotient rule.

17 Note that we can compute the erivatives of all 6 trigonometric functions using the previous slie an the quotient rule. (sin x) = cos x x (cos x) = sin x x (tan x) = (sec x)2 x (cot x) = (csc x)2 x (sec x) = sec x tan x x (csc x) = csc x cot x. x

18 Note that we can compute the erivatives of all 6 trigonometric functions using the previous slie an the quotient rule. (sin x) = cos x x (cos x) = sin x x (tan x) = (sec x)2 x (cot x) = (csc x)2 x (sec x) = sec x tan x x (csc x) = csc x cot x. x As an exercise let s verify a few of these!

19 Computational Examples 1. f (x) = x 2 sin x 2. f (θ) = sin θ 1 + cos θ 3. s(t) = e t sin t 4. y = t sin t 1 + t 2

20 Let y = x + tan x. Fin an equation of the line tangent to the graph of this function at the point (π, π).

21 Let y = x + tan x. Fin an equation of the line tangent to the graph of this function at the point (π, π). Solution: f (x) = 1 + (sec x) 2 an f (π) = 2.

22 Let y = x + tan x. Fin an equation of the line tangent to the graph of this function at the point (π, π). Solution: f (x) = 1 + (sec x) 2 an f (π) = 2. Thus we are looking for a line through the point (π, π) with slope 2.

23 Let y = x + tan x. Fin an equation of the line tangent to the graph of this function at the point (π, π). Solution: f (x) = 1 + (sec x) 2 an f (π) = 2. Thus we are looking for a line through the point (π, π) with slope 2. Such a line is given by y π = 2(x π).

24 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x.

25 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x. Solution: First compute y an y an collect terms in the expression for y + y 2y.

26 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x. Solution: First compute y an y an collect terms in the expression for y + y 2y. Then compare coefficients to get 2 equations involving the unknown values A an B.

27 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x. Solution: First compute y an y an collect terms in the expression for y + y 2y. Then compare coefficients to get 2 equations involving the unknown values A an B. Solve one equation for one of the unknowns in terms of the other one.

28 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x. Solution: First compute y an y an collect terms in the expression for y + y 2y. Then compare coefficients to get 2 equations involving the unknown values A an B. Solve one equation for one of the unknowns in terms of the other one. Substitute into the other equation.

29 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x. Solution: First compute y an y an collect terms in the expression for y + y 2y. Then compare coefficients to get 2 equations involving the unknown values A an B. Solve one equation for one of the unknowns in terms of the other one. Substitute into the other equation. Then you win.

30 Fin constants A an B such that the function y = A sin x + B cos x satisfies the equation y + y 2y = sin x. Solution: First compute y an y an collect terms in the expression for y + y 2y. Then compare coefficients to get 2 equations involving the unknown values A an B. Solve one equation for one of the unknowns in terms of the other one. Substitute into the other equation. Then you win. We fin that A = 3/10 an B = 1/10.