Vocabulary/Definitions
When do we use the chain rule? What is the formula?
Answer: its used when differentiating a 'function of a function' like f(g(x)).
Formula: f'(x)=(g(x)) ⋅ g'(x)
What is f'(6) when f(x)= 3x5 - 2x4 + x3 + 6x2 -3?
f'(x)= 15x4-8x3+3x2+12x
f'(6)= 15(6)4-8(6)3+3(6)2+12(6)
f'(6)= 17,892
Answer: 17,892
Solve ∫(8x3-x2+5x-1)dx
8/4x4-1/3x3+5/2x2-1x+C
2x4-1/3x3+5/2x2-1x+C
Answer: 2x4-1/3x3+5/2x2-1x+C
Find the particular solution of the differential equation which satisfies the given initial condition: dy/dx=3x2-2; y(0)=4
y=∫(3x2-2)dx (0,4)
= x3-x+c ] General solution
4=03-0+C
c=4 Answer: x3-x+4
When do we use the mean value theorem? What is the formula?
Answer: it's used when a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b).
Formula: f'(c)= [f(b)-f(a)]/[b-a]
What is the derivative of f(x)= (x2+6x-2)(3x-7)?
f(x)= (x2+6x-2)(3x-7) = 3x3+18x2-6x-7x2-42x+14
f(x)= 3x3+11x2-48x+14
f'(x)= 9x2+22x-48
Answer: 9x2+22x-48
Solve for ∫01 (x3+2x5+3x10)dx
1/4x4+1/3x6+3/11x11 l01
[14/4+16/3+3(1)11] - 0
[1/4+1/3+3/11]= 113/132
Answer: 113/132
Find the particular solution of the differential equation which satisfies the given initial condition: dy/dx=1/x2: y(1)=4
∫dy/dx=∫(1/x2)dx (1,4)
y=x-1/-1+c = -1/x+c
4=-1/1+c
4=-1 +c Answer: y=-1/x+5
c=5
When do we use the product rule? What is the formula?
Answer: it's used when finding the derivative of a function in which one function is multiplied to another.
Formula: U'V + UV' or f'(x)g(x) + f(x)g'(x)
What is the derivative of f(x)=x2ln(x)?
u=x2 u'=2x
v=ln(x) v'= 1/x
2x(ln(x)) + x2(1/x)
Answer: 2x(ln(x)) + x2(1/x)
Find the solution of ∫(√ x +6x2)dx
∫(x1/2+2x3)dx
1/(3/2)x3/2+2x3+c
2/3x3/2+2x3+c
Answer: 2/3x3/2+2x3+c
What is the difference between a general solution and particular solution?
A general solution included all possible solutions while a particular solution has a specific x or y value.
When do we use the quotient rule? What is the formula?
Answer: it's used when finding the derivative of a function in which one function is being divided by another.
Formula: [f(x)/g(x)]'= [g(x)f'(x)-f(x)g'(x)]/g(x)2
What is the derivative of f(x)= (4x-2)/x2+1?
u=4x-2 u'=4
v=x2+1 v'=2x
f'(x)= 4(x2+1) - 2x(4x-2) / [(x2+1)2]
f'(x)= 4x2+4-8x2+4x / (x2+1)2
Answer: 4x2+4-8x2+4x / (x2+1)2
Solve ∫(2ex+6/x+ln(2))dx
∫2ex +6∫1/x + ln(2)∫dx
2ex + 6lnlxl + (ln2)x + C
Answer: 2ex + 6lnlxl + (ln2)x + C
Find the particular solution of the differential equation which satisfies the given initial condition: dy/dx= -2sin(x); y(π/2)= π
∫dy/dx=∫-2sin(x) (π/2,π)
y=-2(-cosx)+c
y=2cos(x)+c
π=2cos(π/2)+c Answer: 2cos(x)+π
π=2(0)+c c=π
What is the formula for a composite formula?
Formula: (g⋅f)(x)=g(f(x))
Using f(x)=x-4 and g(x)=-3x3+5x, what is g(f(x)) if x=2?
g(f(x))= -3(x-4)3+5(x-4)
g(f(2))= -3(2-4)3+5(2-4)
g(f(2))= -3(-2)3+5(-2)
g(f(2))= 14 Answer: g(f(2))=14
Solve ∫(x3-2x2)((1/x)-5)
∫(x2-5x3-2x+10x2)dx
∫(-5x3+11x2-2x)dx
-5/4x4+11/3x3-x2+C
Answer: -5/4x4+11/3x3-x2+C
Find the particular solution of the differential equation which satisfies the given initial condition: dy/dx= 3/2y; y=4 and x=5
ydy=3/2dx
∫ydy=∫3/2dx
y2/2=3/2x+c
(4)2/2=3/2(4)+c Answer: y2/2= 3/2x+2
8=6+c C=2 *answer can be simplified more
What is Dr.C name?
Fritz (come on guys check your google classroom)