Areas of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of Exercise 77 to match the closed curse with its area. (These exercises were based on “The Surveyor's Area Formula'' by Bart Braden, College Mathematics Journal, September 1986. pp. 335-337, by permission of the author.) (a). 8 3 a b (b). 3 8 π a 2 (c). 2 π a 2 (d). π a b (e). 2 π a b (f). 6 π a 2 Cardiod: ( 0 ≤ t ≤ 2 π ) x = 2 a cos t − a cos 2 t y = 2 a sin t − a sin 2 t
Areas of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of Exercise 77 to match the closed curse with its area. (These exercises were based on “The Surveyor's Area Formula'' by Bart Braden, College Mathematics Journal, September 1986. pp. 335-337, by permission of the author.) (a). 8 3 a b (b). 3 8 π a 2 (c). 2 π a 2 (d). π a b (e). 2 π a b (f). 6 π a 2 Cardiod: ( 0 ≤ t ≤ 2 π ) x = 2 a cos t − a cos 2 t y = 2 a sin t − a sin 2 t
Solution Summary: The author explains how to calculate the area of the curve in the interval ale xl b.
Areas of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of Exercise 77 to match the closed curse with its area. (These exercises were based on “The Surveyor's Area Formula'' by Bart Braden, College Mathematics Journal, September 1986. pp. 335-337, by permission of the author.)
(a).
8
3
a
b
(b).
3
8
π
a
2
(c).
2
π
a
2
(d).
π
a
b
(e).
2
π
a
b
(f).
6
π
a
2
Cardiod:
(
0
≤
t
≤
2
π
)
x
=
2
a
cos
t
−
a
cos
2
t
y
=
2
a
sin
t
−
a
sin
2
t
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.