Rewriting Before Integrating In Exercises 11-14, complete the table to Find the indefinite integral . Original Integral Rewrite Integrate Simplify ∫ 1 x x d x
Rewriting Before Integrating In Exercises 11-14, complete the table to Find the indefinite integral . Original Integral Rewrite Integrate Simplify ∫ 1 x x d x
Solution Summary: The author explains that the integral can be re-written as c
Rewriting Before Integrating In Exercises 11-14, complete the table to Find the indefinite integral.
Original Integral Rewrite Integrate Simplify
∫
1
x
x
d
x
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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).
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