Functions
find the critical numbers for f(x) = x-2lnx
square root(e)
What is f(x), given the following function? What value would you select for "a"?
(8.06)2/3
f(x) = x2/3
8
with which 2 indeterminate forms can you use l'Hospital's rule?
0/0 & infinity/infinity
what are the formulas for volume and surface area of a sphere and cylinder, and volume of a cone?
sphere:
V=4/3*pi*r3 , SA = 4*pi*r2
cylinder:
V= pi*r2*h , SA = pi*r2*h + 2*pi*r2
cone:
V = 1/3*pi*r2*h
list a pro of calculus and a con of calculus (vs using a calculator)
pros: exact values, graphical representation of ideas, perfect viewing window
cons: trigogebra, relying on our drawing abilities
use the first derivative test to determine the local max and min of the function f(x) = x / (x2+4)
BONUS: use the second derivative test to determine the local max and min of the same function
local min: -2
local max: 2
what is the formula for absolute error? relative error?
absolute: dy
relative: dy/y
what is lim as x -> 1 of (6x-3)/(x3+9) ?
3/10
Water spills from a ruptured water tower and spreads in a circular pattern. If the diameter of the water spill increases at a constant rate of 2 m/s, how fast is the area of the spill increasing when the radius is 30m?
60*pi m2/s
in what way is a calculator helpful for solving problems?
when you need to do quick calculations, fast visualizations, and approximations
suppose the derivative of a function is f'(x) = (x+1)2(x-3)5(x-6)4. on what intervals is f increasing?
(3, infinity)
Estimate the number using a linear approximation:
1/1002
0.000998
find the limit as x -> 2 of
(x3-7x2+10x)/(x2+x-6)
-6/5
A street light is mounted at the top of a 15ft tall pole. Prof. Fisher is 6 ft tall and walks away from the pole at 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft away from the pole?
25/3 ft/s
what are the steps of the Mean Value Theorem?
1. Find a Slope
2. Take a derivative
3. Set them equal
Find the absolute extrema for
2x3 − 3x2 − 12x + 1 on the interval [−2, 3]
abs max: 8 (at x = -1)
abs min: -19 (at x = 2)
use linearization to estimate the value of e0.1
approx. 1.1
find the limit as x -> infinity of
(x2+e4x)/(2x-ex)
negative infinity
2 sides of a triangle have fixed lengths of 12 m and 15 m. The angle between them is increasing at a rate of 2o per minute. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60o?
sqroot(7)pi/21 m/min
approx 0.396 m/min
find the values guaranteed by MVT for f(x) = x3+3x2-2 on [-2, 0]
(-3+sqroot(3))/3 , (-3-sqroot(3))/3
determine the concavity and inflection points of the following function:
f(x) = 3x2/3_x
concave down from (-infinity, 0) and (0, infinity)
no inflection points
estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter of 50 m.
5pi/8, or approx 2m3
find the limit as x -> 0+ (cotx - 1/x)
0
a ferris wheel with a radius of 10m is rotating at a rate of one revolution every 2 mins. How fast is a rider rising when her seat is 16m above ground level?
8pi m/min
find the value(s) guaranteed by the MVT for the function f(x) = -(-5x+25)1/2 on [3,5]