AROC/IROC
Consider the function f(x) = 0.02x2 - 1.6x + 20.5 over the interval [25, 35]. Calculate the average rate of change.
AROC= -4.5
Let f and g be the functions defined by f(x) = -0.5x3 + 3x + 1 and g(x) = ex/2. Let h be the vertical distance between the graphs of f and g for the interval [0, 2]. Find the rate at which h changes with respect to x when x = 1.5.
h' (1.5) = -1.4335
The following table lists the values of functions f and g, and of their derivatives, f' and g', for x = 4.
x= 4
f(x)= -4
g(x)= 13
f'(x)= 0
g'(x)= 8
Let function F be defined as F(x)= f(x)/g(x)
F'(4) =
F'(4) = 8
(Calculator Question)
The derivative of the function f is given by
f'(x) = x^2 - 2 - 3x(cos(x))
On which of the following intervals in [-4, 3] is f decreasing?
A) [-4, -3.444], [-1.806, -0.660], and [1.509, 3]
B) [-4, -2.805] and [-1.227, 0.637]
C) [-3.444, -1.806] and [-0.660, 1.509]
D) [-2.805, -1.227] and [0.637, 3]
C
What is the derivative of cot (x)?
-csc2 (x)
Consider the function f(x)=x^2. Find the instantaneous rate of change at x=2.
IROC=4
Use the table to estimate the derivative at the given point.
x hours 1 3 4 7 9
v(x) visitors 120 476 595 807 902
f'(8) =
f' (8) = 47.5 visitors/hour
The following table lists the values of the functions f and g, and of their derivatives, f' and g', for x=4.
x= 4
f(x)= -4
g(x)= 13
f'(x)= 0
g'(x)= 8
Let function F be defined as F'(x) = f(x)g(x)
F'(4) =
F'(4) = -32
Let g(x)=x^3-16x and let c be the number that satisfies the Mean Value Theorem for g on the interval [-4,2].
What is c?
c = -2
What is the derivative of f(x) = tan (x)?
f'(x) = sec2 (x)
Find the slope of the tangent line to y= Ax2 - 4 at x=-3, where A is a constant.
m=IROC= -6A
Write the equation of the tangent line at the given value of x. You may use a calculator.
f(x) = ln 2x / 4x at x = 1
y - 0.173 = 0.0767 (x - 1)
What is the derivative of the following?
f(x) = (6x3) / ln (x)
f'(x) = (18x2ln (x) - 6x2) / (ln(x))2
Determine if the Mean Value Theorem can be applied to the following function on the given closed interval. If so, find all possible values of c. f(x) = 3 + square root of x on [0, 4].
c = 1
What is the derivative of f(x) = 2tan(x) - 3cot(x)?
f'(x) = 2sec2x + 3csc2x
If a rock is thrown upward on Mars, its height (in meters) after t seconds is given by s(t) = 16t - 1.86t2. At what time is the instantaneous rate of change of the rock equal to -2.6 m/s?
*Use one of the instantaneous rates of change formulas
t=5
The graph of y = 3 - e5x crosses the x-axis at one point. What is the slope of the graph at this point?
y'(a)= -15
Find the derivative of the given function:
f(x) = x7 sin (sin-1 (x))
f'(x) = 8x7
Let f be a function defined for all real numbers except for 0.
Also let f', the derivative of f, be defined as
f'(x)= ((x-2)^3)/x.
On which intervals is f increasing?
(-infinity, 0) and (2, infinity)
Find the derivative of f(x) = csc x + x tan x
f'(x) = -csc x cot x + tan x + x sec2 x
The wind chill is the temp, in F, a human feels based on the air temperature, in F, and the wind velocity v, in mph. If the air temp is 32 degrees F, then the wind chill is given by W(v) = 55.6-22.1v^0.16 and is valid for the interval [5, 60]
Find the average rate of change of W over the interval [5, 60]. Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval [5, 60].
Average rate of change of W over the interval [5, 60] is = -0.254
W'(v) = the AROC of W when v = 23.011
Find the equation of a line tangent to the graph of f(x) = cot (x) at x = pi/4
y - 1 = -2(x - pi/4)
What is the derivative of f(x)?
f(x) = sin (x) / tan (x)
f'(x) = -sin (x)
Let s(t) be the total miles that you have driven on your trip down a toll road freeway after time t hours. Assume also that s(1/3)=10 miles and s(1.2)=82 miles. At a checkpoint along the toll road you admit these facts to a highway patrol officer, who is familiar with the Mean Value Theorem. The officer makes a few quick calculations, smiles, then politely prepares to write you a traffic ticket. Explain why.
s'(c) = 1080/13. approx 83.1 mph.
You must have been speeding at t = c hours.
What is the derivative of f(x) = tan x at x = pi/6?
f'(pi/6) = 4/3