Category 1: Derivatives of Exponentials: eu
Directions: Look at the examples below and answer the question.
1) Describe in your own words what to do to find the derivative for problems like those above.
...
Category 1: Derivatives of Exponentials: eu
Directions: Look at the examples below and answer the question.
1) Describe in your own words what to do to find the derivative for problems like those above.
Derivatives of Exponentials: au
Directions: Look at the examples below and answer the question.
2) Describe in your own words what to do to find the derivative for problems like those above
3) Find the derivative; show all work: f(x) = 57x
4) Find the derivative, show all work: f ( x)=59x2+6 x
5) If f(x) = 4xe3x, what previously learned rule must be used to find its derivative? Find the derivative of f(x). Simplify and factor your answer.
2e4 x
6) If f ( x)= 3 sin(5 x) , what previously learned rule should be used to find its derivative?
Find f ’(x). Show all work. If you use Wolfram, submit screen shot here.
7) If y=arctan (e x4 ) Find dy
. If you use Wolfram, submit screen shot here.
Category 2: Derivatives of Logarithms
Directions: Look at the examples below and answer questions 8 and 9.
hifs(xstu)d=y solnur(ce3wxa)s downloaded bfy(x) = ln x2. from CourseHer
100000820529148 of.c(oxm) o=n ln(x3 + 045):40:29 GMT
04-10-2021 f-(0x5:)00= ln(sin(2x))
f ’(x) = 1
x f ’(x) = 2
x f ’ (x) = 3 x2
x3+ 4 f ’ (x) =
2 cos(2 x )
sin(2 x ) =2 cot(2 x)
8) Describe in your own words what to do to find the derivative for problems like those above.
9) y = 4 ln(3x2), find dy/dx. Show all work.
Logarithmic Expansions
Just as simplifying trig expressions makes them easier to differentiate, expanding or collecting logarithms can make differentiation easier, too.
Rules of logarithmic expansions:
1) ln(ab) = ln a + ln b 2) ln (a/b) = ln a – ln b 3) ln ab = b ln a
10) For each function expansion below, list the number of the rule(s) above (1, 2, or 3) that were used to expand the expression.
A) Function: f ( x )=ln ( x √x +1) Expansion: f ( x )=ln( x )+ 1 ln ( x+1)
Rule(s):
B) Function: f ( x )=ln(
Rule(s):
x2
x −1
) Expansion: f ( x )=2 ln ( x )−ln ( x−1)
C) Function: f ( x )=ln √x Expansion:
xcsc ( x)
Rule(s):
csc ( x)
f ( x )= 1 ln ( x )−ln x−ln ¿ )
2
11) Expand and find the derivative: f ( x)=ln x2 ex
2 x−9
If you use Wolfram, show screen shot.
Category 3: L’Hopital’s Rule: A new way to find an old friend: the limit. Directions: Look at the examples below and answer questions 12 and 13.
lim ex−1 lim ex
x →0 → 0 ⇒ x→ 0 =e0=1
x 0 1
lim cos ( x)−1 0 lim −sin ( x) 0
x →0 → ⇒ x →0 = =0
2 sin( x ) 0 2cos ( x) 2
lim ex lim ex lim ex lim ex
x→ ∞ → ∞ ⇒ x →∞ → ∞ ⇒ x → ∞ → ∞ ⇒ x→∞ =∞
x3+2 x2 ∞ 3 x2+4 x ∞ 6 x + 4 ∞ 6
12) What steps were taken using L’Hopital’s rule to find the limits above?
13) Can L’Hopital’s rule be used to find any limit? Explain.
14) Find the limit using L’Hopital’s rule:
lim 2 ex−1+3
x →1
ln( x)
. Show all work.
15) Find the limit using L’Hopital’s rule:
f(x) e^x-e^-x= 1-(-1)=2/0 g(x) 4sin(x)=4
f'(x) e^-x-e^x= -1+1=0/0 g'(x) 4cos(x)=0
lim ex−e−x
x →0
4 sin ( x)
. Show all work.
lim sin(x)
16) Find the limit:
x0 x . Show all work. Wolfram OK.
f(x)= sin(x)/g(x)=0
f'(x)=cos(x)/g(x)=1
=0/1=0
Part II: Use Wolfram or your Calculator to determine the following: you don’t need to use screen shots, but do show your work and indicate how you made your calculations: e.g, after a calculation you could write “[Wolfram]” or “ [TI-89]”
17) The slope of the tangent to the curve f ( x )=√ x−ln( x3+5) at the point (4, -2.234).
18 Evaluate the derivative of f ( x)= 1 −ex3−2 x +4 at x = 1.
x4
19) A stunt driver plans to drive his specially equipped motorcycle up a curved ramp and become airborne in order to cross a ravine and land on the other side.
Being a great mathematician, the driver concluded that the motorcycle will follow the path given by the following piecewise function: t in seconds, f(t) in ft.
xmin = -3, xmax = 25, y-min = -10, y-max = 35
a.) What is the velocity of the motorcycle at 10 seconds, 14 seconds, and 18 seconds? b.) What is the acceleration at 10 seconds, 14 seconds, and 18 seconds?
c.) How long will it take for the motorcycle to reach the top of the path? d.) How long will it take to hit the ground on the other side of the ravine?
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