In each part, match the contour plot with one of the functions f ( x , y ) = x 2 + y 2 , f ( x , y ) = x 2 + y 2 , f ( x , y ) = 1 − x 2 − y 2 by inspection, and explain your reasoning. Larger values of z are indicated by lighter colors in the contour plot, and the concentric contours correspond to equally spaced values of z.
In each part, match the contour plot with one of the functions f ( x , y ) = x 2 + y 2 , f ( x , y ) = x 2 + y 2 , f ( x , y ) = 1 − x 2 − y 2 by inspection, and explain your reasoning. Larger values of z are indicated by lighter colors in the contour plot, and the concentric contours correspond to equally spaced values of z.
In each part, match the contour plot with one of the functions
f
(
x
,
y
)
=
x
2
+
y
2
,
f
(
x
,
y
)
=
x
2
+
y
2
,
f
(
x
,
y
)
=
1
−
x
2
−
y
2
by inspection, and explain your reasoning. Larger values of z are indicated by lighter colors in the contour plot, and the concentric contours correspond to equally spaced values of z.
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).
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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