Download College Algebra/Trig Summer Workshop Trigonometric Derivatives Name:

College Algebra/Trig Summer Workshop
Trigonometric Derivatives
Name:
Find all of the points where f (x) = x + sin x has a horizontal tangent line
(recall that f 0 (x) gives the slope of the tangent line at x)
(hint: f 0 (x) = 1 + cos x)
Find the equation of the tangent line to the curve f (x) = (sin x + cos x) sec x at (π, 1)
You are going to differentiate x4 + sin y = x3 y 2 using implicit differentiation. After performing the
dy
dy
dy
implicit differentiation you get 4x3 + cos y dx
= 3x2 y 2 + 2yx3 dx
. Solve this equation for dx
.
Class Examples
a) Find the equation of the tangent line to the curve f (x) = 1 + 2 sin x at x =
(hint: f 0 (x) = 2 cos x)
π
.
6
b) Find all the points where the graph f (x) = x − cot x has a horizontal tangent line
(hint: f 0 (x) = 1 + csc2 x)
c) Find the critical points of the function f (x) = x + 2 cos x
(hint: f 0 (x) = 1 + 2 sin x)
d) If the position of a spring is given by s(t) = 5 cos t then the velocity of the spring is s0 (t) = v(t) =
−5 sin t. Find all times on the interval [0, 2π] when the velocity is positive.
e) You are going to differentiate x sin(2y) = y cos(2x) using implicit differentiation. After performing the
dy
dy
= cos(2x) dx
− 2y sin(2x). Solve this equation for
implicit differentiation you get sin(2y) + 2x cos(2y) dx
dy
.
dx
Homework Problems
1) Find the equation of the tangent line to the curve f (x) = 3x3 + sin x at x = 0.
(hint: f 0 (x) = 9x2 + cos x)
2) Find the equation of the tangent line to the curve f (x) = 4 sin x cos x at x =
(hint: f 0 (x) = 4 (cos x cos x + sin x(− sin x)))
π
.
3
3) Find the critical points of the function f (x) = e6x cos x
(hint: f 0 (x) = 6e6x cos x + e6x (− sin x))
4) Find all the points where the graph f (x) = 2x + sin x has a horizontal tangent line
(hint: f 0 (x) = 2 − cos x)
5) You are going to differentiate y sin y = 1 − xy using
implicit
differentiation. After performing the
dy
dy
dy
dy
.
implicit differentiation you get sin y dx − y cos y dx = − y + x dx . Solve this equation for dx