Radioactive Decay A sample of radioactive material decays over time (measured in hours) with decay constant .2 . The graph of the exponential function y = P ( t ) in Fig.7 gives the number of grams remaining after t hour [Hint: In parts (c) and (d) use the differential equation satisfied by P ( t ) .] a. How much was remaining after 1 hour? b. Approximate the half-life of the material. c. How fast was the sample decaying after 6 hours? d. When was the sample decaying at the rate of . 4 grams per hour?
Radioactive Decay A sample of radioactive material decays over time (measured in hours) with decay constant .2 . The graph of the exponential function y = P ( t ) in Fig.7 gives the number of grams remaining after t hour [Hint: In parts (c) and (d) use the differential equation satisfied by P ( t ) .] a. How much was remaining after 1 hour? b. Approximate the half-life of the material. c. How fast was the sample decaying after 6 hours? d. When was the sample decaying at the rate of . 4 grams per hour?
Solution Summary: The author calculates the remaining sample of a radioactive material after 1 hour. The graph of the exponential function y=P(t) gives the number of grams remaining.
Radioactive Decay A sample of radioactive material decays over time (measured in hours) with decay constant
.2
. The graph of the exponential function
y
=
P
(
t
)
in Fig.7 gives the number of grams remaining after
t
hour [Hint: In parts (c) and (d) use the differential equation satisfied by
P
(
t
)
.]
a. How much was remaining after
1
hour?
b. Approximate the half-life of the material.
c. How fast was the sample decaying after
6
hours?
d. When was the sample decaying at the rate of
.
4
grams per hour?
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.
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY