Given r x = − x 2 , s x = cot x , and t x = csc x , a. Find r ∘ t x b. Graph s x and r ∘ t x together on the ZTRIG window. The relationship between the two graphs will be studied in calculus. For a given value of x in the domain of s x = cot x , y = − csc 2 x gives the slope of a line tangent to s at x .
Given r x = − x 2 , s x = cot x , and t x = csc x , a. Find r ∘ t x b. Graph s x and r ∘ t x together on the ZTRIG window. The relationship between the two graphs will be studied in calculus. For a given value of x in the domain of s x = cot x , y = − csc 2 x gives the slope of a line tangent to s at x .
Given
r
x
=
−
x
2
,
s
x
=
cot
x
,
and
t
x
=
csc
x
,
a. Find
r
∘
t
x
b. Graph
s
x
and
r
∘
t
x
together on the ZTRIG window. The relationship between the two graphs will be studied in calculus. For a given value of
x
in the domain of
s
x
=
cot
x
,
y
=
−
csc
2
x
gives the slope of a line tangent to
s
at
x
.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.