3.4.1 Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1. ...
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3.4.2 Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1. ...
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3.4.3 Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1. ...
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3.4.4 Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1. ...
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3.4.5 Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1. ... (you will need to check an intermediate point).
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3.4.6 Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1. ... (you will need to check an intermediate point).
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3.4.7 Use the Extreme Value Theorem to show that each of the following functions has a positive maximum on the interval 0≤ x ≤1. ...
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3.4.8 Use the Extreme Value Theorem to show that each of the following functions has a positive maximum on the interval 0≤ x ≤1. ...
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3.4.9 Use the Extreme Value Theorem to show that each of the following functions has a positive maximum on the interval 0≤ x ≤1. f ( x )=5x(1 − x)(2 − x) − 1.
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3.4.10 Use the Extreme Value Theorem to show that each of the following functions has a positive maximum on the interval 0≤ x ≤1. ...
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3.4.11 Find the points guaranteed by the Mean Value Theorem and sketch the associated graph. The slope of the function f (x) = ... must match the slope of the secant connecting x = 0 and x = 1.
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3.4.12 Find the points guaranteed by the Mean Value Theorem and sketch the associated graph. The slope of the function f (x) = ... must match the slope of the secant connecting x = 0 and x = 2.
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3.4.13 Find the points guaranteed by the Mean Value Theorem and sketch the associated graph. The slope of the function g(x)= ...must match the slope of the secant connecting x = 0 and x = 1.
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3.4.14 Find the points guaranteed by the Mean Value Theorem and sketch the associated graph. The slope of the function g(x)= ...must match the slope of the secant connecting x = 0 and x = 2.
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3.4.15 Draw functions with the following properties. A function with a global minimum and global maximum between the endpoints.
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3.4.17 Draw functions with the following properties. A
differentiable function with a global maximum at the left endpoint, a
global minimum at the right endpoint, and no critical points.
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3.4.17 Draw functions with the following properties. A
differentiable function with a global maximum at the left endpoint, a
global minimum at the right endpoint, and no critical points.
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3.4.18
Draw functions with the following properties. A function with a
global maximum at the left endpoint, a global minimum at the right
endpoint, and at least one critical point.
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3.4.19
Draw functions with the following properties. A function with a
global minimum and global maximum between the endpoints, but no critical
points.
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3.4.20 Draw functions with the following properties. A function that never reaches a global maximum.
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3.4.21
Check
whether the Intermediate Value Theorem and Mean Value Theorem fail in
the following cases where the function is not continuous. Consider the
Heaviside function (Section 2.3, Exercise 25) defined by ...
Show that there is no solution to the equation H (x) = 0.5, and that there is no tangent that matches the slope of the secant connecting x =−1 and x = 1.
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3.4.22 Check
whether the Intermediate Value Theorem and Mean Value Theorem fail in
the following cases where the function is not continuous. Consider the absolute value function g(x) = |x|.
Does this satisfy the conditions for the Intermediate Value Theorem?
Show that there is no tangent that matches the slope of the secant
connecting x =−1 and x = 2.
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3.4.23 There is a clever proof of the Mean Value Theorem from Rolle’s Theorem. The idea is to tilt the function f so that it takes on the same values at the endpoints a and b. In particular, we apply Rolle’s Theorem to the function\ ... For the following functions, show that g(a) = g(b), apply Rolle’s Theorem to g, and find the derivative of f at a point where g ′ (x)=0. f ( x )= ..., a = 1, and b = 2.
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3.4.24 There is a clever proof of the Mean Value Theorem from Rolle’s Theorem. The idea is to tilt the function f so that it takes on the same values at the endpoints a and b. In particular, we apply Rolle’s Theorem to the function ... For the following functions, show that g(a) = g(b), apply Rolle’s Theorem to g, and find the derivative of f at a point where g ′ (x)=0. In general, without assuming a particular form for f (x) or values for a and b.
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3.4.25
Try to apply the Intermediate Value Theorem to the following
problems. The price of gasoline rises from $1.199 to $1.279. Why is it
not necessarily true that the price was exactly $1.25 at some time?
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3.4.26 Try to apply the Intermediate Value Theorem to the following problems. A
pot is dropped from the top of a 500-ft building exactly 200 ft above
your office. Must it have fallen right past your office window?
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3.4.27 Try to apply the Intermediate Value Theorem to the following problems. A cell takes up 1.5 ×...milliliters of water in the course of an hour. Must the cell have taken up exactly 1.0 ×...milliliters at some time? Is it possible that the cell took up exactly 2.0 ×...milliliters at some time?
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3.4.28 Try to apply the Intermediate Value Theorem to the following problems. The
population of bears in Yellowstone Park has increased from 100 to 1000.
Must it have been exactly 314 at some time? What additional assumption
would guarantee this?
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3.4.29
The Intermediate Value Theorem can often be used to prove that
complicated discrete-time dynamical systems have equilibria. Prove
that the discrete-time dynamical system ... has an equilibrium between 0
and ...
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3.4.30
The Intermediate Value Theorem can often be used to prove that
complicated discrete-time dynamical systems have equilibria. A lung
follows the discrete-time dynamical system ... ... where γ =5.0. Show that there is an equilibrium between 0 and γ .
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3.4.31
The Intermediate Value Theorem can often be used to prove that
complicated discrete-time dynamical systems have equilibria. lung
follows the discrete-time dynamical system ... ... where γ = 5.0 and the function α(ct ) is positive, decreasing, and α(0)= 1. Show that there is an equilibrium between 0 and γ .
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3.4.32
The Intermediate Value Theorem can often be used to prove that
complicated discrete-time dynamical systems have equilibria. A lung
follows the discrete-time dynamical system ... ...We know only that neither ...can
exceed 1 mole/liter. Use the Intermediate Value Theorem to show that
this discrete-time dynamical system must have an equilibrium.
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3.4.33 The Intermediate Value Theorem has applications in agricultural transport. A
farmer sets off on Saturday morning at 6 a.m. to bring a crop to
market, arriving in town at noon. On Sunday she sets off in the opposite
direction at 6 a.m. and returns home along the same route, arriving
once again at noon. Use the Intermediate Value Theorem to show that at
some point along the path, her watch must read exactly the same time on
the two days.
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3.4.35
An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is a
differentiable function of time. Why must the mass have been exactly
10 kg at some time?
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3.4.35
An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is a
differentiable function of time. Why must the mass have been exactly
10 kg at some time?
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3.4.36
An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is a
differentiable function of time. Why must the rate of increase have
been exactly 4.0 kg/yr at some time?
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3.4.37 An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is a differentiable function of time. Draw
a graph of mass against time where the mass is increasing, is equal to
10.0 kg at 13 yr, and has a growth rate of exactly 4.0 kg/yr after 1 yr.
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3.4.38 An organism grows from 4 kg to 60 kg in 14 yr. Suppose that mass is a differentiable function of time. Draw
a graph of mass against time where the organism reaches 10.0 kg at 1
yr, and has a growth rate of exactly 4.0 kg/yr at 13 yr.
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3.4.39 Draw
the positions of cars from the following descriptions of 1-h trips.
What speed must the car achieve according to the Mean Value Theorem?
What speeds must the car achieve according to the Intermediate Value
Theorem? A car starts at 60 mph and slows down to 0 mph. The average speed is 20 mph after 1 h.
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3.4.40 Draw
the positions of cars from the following descriptions of 1-h trips.
What speed must the car achieve according to the Mean Value Theorem?
What speeds must the car achieve according to the Intermediate Value
Theorem? A car starts at 60 mph, steadily slows down to 20 mph,
and then speeds up to 50 mph by the end of 1 h. The average speed over
the whole time is 40 mph.
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3.4.41 Draw
the positions of cars from the following descriptions of 1-h trips.
What speed must the car achieve according to the Mean Value Theorem?
What speeds must the car achieve according to the Intermediate Value
Theorem? A car drives 60 miles in 1 h, and never varies speed by more than 10 mph.
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3.4.42 Draw
the positions of cars from the following descriptions of 1-h trips.
What speed must the car achieve according to the Mean Value Theorem?
What speeds must the car achieve according to the Intermediate Value
Theorem? In a test, a car drives zero net distance in 1 h by
switching from reverse to forward at some point. The test includes
achieving the maximum possible reverse speed (20 mph) and the maximum
possible forward speed (120 mph).
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3.4.43 The Marginal Value Theorem states that the best ... where τ is the travel time to the next cabbage patch and F(t) is the total amount of food gathered in one location in time t. Suppose that τ =1 and ... Sketch the associated figure (as in Figure 3.3.14) and estimate the solution.
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3.4.44 The Marginal Value Theorem states that the best ... where τ is the travel time to the next cabbage patch and F(t) is the total amount of food gathered in one location in time t. Suppose that τ =1 and ... Use the Intermediate Value Theorem to prove that there is a solution.
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3.5.1 Find the following limits. ...
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