Definitions
What are other terms that describe the derivative? (Name two)
Instantaneous rate of change and slope of tangent line
Find the derivative of f(x)= 3x+ (1/x) ? Give f'(x) and state derivative rule used.
Power Rule
f'(x)= 3- (1/x2)
Check both signs of acceleration and velocity
lim x->0 (3x2+7)/(x-2)
7/2
If f(x) is continuous and f(3)=-4 and f(10)= 12, does there exist f(c)= 5? Justify.
Since f(x) is continuous, we can apply the IVT. By the IVT, the range is [-4,12]. Thus, 5 falls in between. There exists f(c)=5.
A line that passes through the curve at x=a and has a slope m=-1/F '(a). Name the type of line.
Normal Line
Find the derivative of f(x)= ex(x2+4x)? Give f'(x) and state derivative rule used.
Product Rule
f'(x)= ex(2x+4)+ ex(x2+4x
If v(t)= 3t2-12t+9, find the average acceleration over the interval [1,2].
lim x-> 4 (x2-8x+16)/(x-4)
Must show all steps and explain how solve.
0
What are examples of non-differentiable types? Name all 4.
corner, cusps, vertical tangent lines, discontinuities (jump, remove and infinite)
When do you use the chain rule?
Composite functions. A function inside another.
If(x)= e2x and Johnny says the f'(x)=2ex, what is his mistake?
Chain rule incorrectly used. The original inside of 2x must be kept in the exponent when taken the derivative.
The position of a particle is given by s(t)= t3-6t2+9t. Is the particle speeding up or slowing down at t= 4? Justify.
Speeding up because v(4)= 9 and a(4)= 12. Both v(4) and a(4) have the same sign.
lim x-> (ln x)/(x-1)
Must show all steps and explain how solve.
=1
What types of discontinuities do rational functions have? Where occur?
Removable at holes and infinite at vertical asymptotes.
What are the 5 key steps for related rates?
1. Annotate
2. Identify governing equation that relates all quantities and rates.
3.Differentiate with respect to time
4. Substitute givens
5. Solve for missing rate
Find the derivative of f(x)= csc(x4 + ln x) ? Give f'(x) and state derivative rule used.
Chain Rule
f'(x)= -csc(x4+lnx)cot(x4+lnx)(4x3+ (1/x))
The position of a particle is given by s(t)= t3-6t2+9t. What is one interval over which the particle is slowing down? Must show sign chart for v(t) and a(t)
(0,1)
or
(2,3)
lim x->0 (sin x - x)/(x2)
Must show all steps and explain how solve.
0
What must you ensure to do when using L'Hospital's Rule? (2 main things)
1. Prove indeterminate form
2. Use double limit notation (numerator and denominator) when doing direct substitution.
What is the different between constraint and optimization equation?
Constraint equation is equal to a value (value given in the prompt) and optimization is the equation the question is asking to maximize or minimize (unknown value).
If f'(x)= 3x2+ ex, what is f(x)?
f(x)= x3+ex
Given v(t)=5x2+5x-60, when does the particle change direction? Justify.
t=3 because v(t)=0 and changes sign.
lim x->3 (x+tan x)/ (sin x)
Must show all steps and explain how solve.
2
If f(9)= 10 and f'(9)= -2, approximate f(9.2) using a tangent line approximation.
y-10=-2(x-9)
f(9.2) is about 9.6