1. Problem of the Day
(Calculator Allowed)
If y = 2x - 8, what is the minimum value of the product xy?
a) b) -8
c) d) 0
e) 2

2. Problem of the Day
(Calculator Allowed)
If y = 2x - 8, what is the minimum value of the product xy?
a) b) -8
c) d) 0
e) 2
xy = x(2x - 8)
= 2x2 - 8x
Graph and find minimum

3. Implicit Differentiation
Can we still determine the slope of the tangent line to the graph of x2 + 4y2 = 4 at the point (√2, -1/√2)?

4. Differentiate 2x + 8y dy = 0 dx dy = -2x = -x dx 8y 4y
Implicit Differentiation
Can we still determine the slope of the tangent line to the graph of x2 + 4y2 = 4 at the point (√2, -1/√2)?
Differentiate
2x + 8y dy = 0 dx
Solve for dy/dx
dy = -2x = -x
dx 8y 4y

5. Differentiate 2x + 8y dy = 0 dx dy = -2x = -x dx 8y 4y
Implicit Differentiation
Can we still determine the slope of the tangent line to the graph of x2 + 4y2 = 4 at the point (√2, -1/√2)?
Differentiate
2x + 8y dy = 0 dx
Solve for dy/dx
dy = -2x = -x
dx 8y 4y
Substitute at point
dy = -√2 = 1 dx √2

6. Given sin y = x, find the largest interval that is
differentiable

7. Given sin y = x, find the largest interval that is differentiable
d [sin y] = dx
dx dx
since cos y cannot be zero
when would it be zero?
cos y dy = 1 dx
dy = 1
dx cos y

8. Given sin y = x, find the largest interval that is differentiable
d [sin y] = dx
dx dx
cos y cannot be zero
when would it be zero?
cos y dy = 1 dx
at Π/2 and -Π/2
dy = 1
dx cos y

9. Given sin y = x, find the largest interval that is differentiable
d [sin y] = dx
dx dx
cos y dy = 1 dx
can the function be written as a function of x?
dy = 1
dx cos y

10. Given sin y = x, find the largest interval that is differentiable
d [sin y] = dx
dx dx
can the function be written as a function of x?
cos y dy = 1 dx
dy = 1
dx cos y
sin2y + cos2y = 1
cos y = √1- sin2y
From original equation we know that sin y = x so
dy = Π/2 < x < Π/2
dx √1- x2

11. Second Derivative Implicitly
x2 + y2 = find d2y
dx2
First derivative

12. Second Derivative Implicitly
x2 + y2 = find d2y
dx2
First derivative
x2 + y2 = 25
2x + 2y dy = 0 dx
dy = -2x = -x
dx y y
Second derivative

13. . Second Derivative Implicitly dy = -2x = -x x2 + y2 = 25 find d2y
First derivative
dy = -2x = -x
dx y y
x2 + y2 = find d2y
dx2
Second derivative
d2y = y(-1) - (-x) dy/dx
dx y2
quotient rule
= -y + x(-x/y) y2
substitute for dy/dx
find common denominator
for numerator and rewrite as multiplication .
= -y2 - x
y y2
= -(y2 + x2) y3
= -25 y3
substitute for x2 + y2

14. Differentiate x + y = 8 with respect to z.
Implicit Differentiation
Remember that you can differentiate with respect to any variable.
Differentiate x + y = 8 with respect to z.
Differentiate x cos x = y with respect to y.
Differentiate x3 + 4z - 2y = 16 with respect to z.

15. Differentiate x + y = 8 with respect to z.
Implicit Differentiation
Remember that you can differentiate with respect to any variable.
Differentiate x + y = 8 with respect to z.
dx + dy = 0
dz dz
Differentiate x cos x = y with respect to y.
x(-sin x dx) + cos x (dx) = 1
dy dy
Differentiate x3 + 4z - 2y = 16 with respect to z.
3x2 dx dy = 0
dz dz

16. Attachments