2.10.1 Find the derivatives of the following functions. ...
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2.10.2 Find the derivatives of the following functions. ...
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2.10.3 Find the derivatives of the following functions. h ( x ) = sin(θ) cos(θ).
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2.10.4 Find the derivatives of the following functions. ...
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2.10.5 Find the derivatives of the following functions. F ( z )=3 + cos(2z − 1).
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2.10.6 Find the derivatives of the following functions. ...
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2.10.7 Find the derivatives of the following functions. ...
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2.10.8 Find the derivatives of the following functions. ...
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2.10.9 Use the definitions and the derivatives of sin(θ) and cos(θ) to check the derivatives of the other trigonometric functions, and to find their second derivatives. tan(θ)
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2.10.10 Use the definitions and the derivatives of sin(θ) and cos(θ) to check the derivatives of the other trigonometric functions, and to find their second derivatives. cot(θ)
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2.10.11 Use the definitions and the derivatives of sin(θ) and cos(θ) to check the derivatives of the other trigonometric functions, and to find their second derivatives. sec(θ)
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2.10.12 Use the definitions and the derivatives of sin(θ) and cos(θ) to check the derivatives of the other trigonometric functions, and to find their second derivatives. csc(θ)
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2.10.13 Use
the angle addition formulas (Equations 2.10.1 and 2.10.2) to expand the
following and use the product and sum rules to compute the derivatives
of the following functions. Compare the result with what you get with
the chain rule. cos(2θ). Simplify the answer in terms of sin(2θ).
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2.10.14 Use
the angle addition formulas (Equations 2.10.1 and 2.10.2) to expand the
following and use the product and sum rules to compute the derivatives
of the following functions. Compare the result with what you get with
the chain rule. sin(2θ). Simplify the answer in terms of cos(2θ).
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2.10.15 Use
the angle addition formulas (Equations 2.10.1 and 2.10.2) to expand the
following and use the product and sum rules to compute the derivatives
of the following functions. Compare the result with what you get with
the chain rule. Take the derivative of cos(θ + φ) with respect to θ, thinking of φ as a constant. Simplify the answer in terms of sin(θ + φ).
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2.10.16 Use
the angle addition formulas (Equations 2.10.1 and 2.10.2) to expand the
following and use the product and sum rules to compute the derivatives
of the following functions. Compare the result with what you get with
the chain rule. sin(θ + φ). Simplify the answer in terms of cos(θ + φ).
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3.3.17
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 5) for −1 ≤ z ≤ 1.
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2.10.18 Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the behavior in words? b ( y ) = ... + 3 cos(y) for 0 ≤ y ≤2π.
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2.10.19 Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the behavior in words? c ( z )= ...sin(z) for 0 ≤ z ≤2π.
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2.10.20 Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the behavior in words? r ( t )=t cos(t) for 0≤t ≤4π.
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2.10.21 Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the behavior in words? s ( t ) = ...cos(t) for 0 ≤ t ≤ 40.
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2.10.22 Compute the derivatives of the following functions. Find the value of the function and the slope at 0, π/2, and π. Sketch a graph of the function on the given domain. How would you describe the behavior in words? p ( t )= ... (1 + 0.2 cos(t)) for 0 ≤ t ≤40.
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2.10.23
We can use Theorem 2.13 to find the derivatives of the inverse
trigonometric functions. Find the derivative of ...If you use the
identity ..., you can write the answer without any trigonometric
functions. Reference Theorem 2.13 ...
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2.10.24
We can use Theorem 2.13 to find the derivatives of the inverse
trigonometric functions. Find the derivative of ... . Write the answer
without any trigonometric functions. Reference Theorem 2.13 ...
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2.10.25
We can use Theorem 2.13 to find the derivatives of the inverse
trigonometric functions. Find the derivative of ...If you use the
identity ...you can write the answer without any trigonometric
functions. Reference Theorem 2.13 ...
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2.10.26
We can use Theorem 2.13 to find the derivatives of the inverse
trigonometric functions. Find the derivative of ...Write the answer
without any trigonometric functions. Reference Theorem 2.13 ...
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2.10.27 Show that the following are solutions of the given differential equation. ...
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2.10.28 Show that the following are solutions of the given differential equation. ...
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2.10.29 Show that the following are solutions of the given differential equation. ...
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2.10.30 Show that the following are solutions of the given differential equation. ...
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2.10.31 Find
the derivatives of the following functions (from Section 1.8, Exercises
35-38). Sketch a graph and check that your derivative has the correct
sign when the argument is equal to 0. ...
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2.10.32 Find
the derivatives of the following functions (from Section 1.8, Exercises
35-38). Sketch a graph and check that your derivative has the correct
sign when the argument is equal to 0. g ( t ) = 4.0 + 3.0 cos(2π(t − 5.0)).
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2.10.33 Find
the derivatives of the following functions (from Section 1.8, Exercises
35-38). Sketch a graph and check that your derivative has the correct
sign when the argument is equal to 0. ...
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2.10.34 Find
the derivatives of the following functions (from Section 1.8, Exercises
35-38). Sketch a graph and check that your derivative has the correct
sign when the argument is equal to 0. ...
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2.10.35 Consider the function ...where T is a constant. Consider a spring with k =0.1 and m =1.0. Find the period T that produces a solution of the spring equation. Is this spring stronger or weaker than one with k = 1.0, and does the oscillation have a larger or smaller period?
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2.10.36 Consider the function ...where T is a constant. Consider a spring with k = 1.0 and m = 5.0. Find the period T that produces a solution of the spring equation. Does a heavier object oscillate more slowly than a light one?
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2.10.37 Consider the combination of the temperature cycles (Subsection 1.8.3) ... given by ... Find the derivative of ...
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2.10.38
Consider the combination of the temperature cycles (Subsection 1.8.3)
... given by ... Sketch a graph of the derivative over one
month. If you measured only the derivative, which oscillation would you
see?
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2.10.39 The
spring we studied had no friction. Friction acts as a force much like
the spring itself, but is proportional to velocity rather than
displacement. One possible equation describing this is ... Explain each term in this equation and show that x(t)= ...is a solution.
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2.10.40
The
spring we studied had no friction. Friction acts as a force much like
the spring itself, but is proportional to velocity rather than
displacement. One possible equation describing this is ... Graph
the solution and explain what is going on. Is friction strong in this
system?
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2.10.41
This problem requires a computer program with a built-in ability to
solve differential equations. The spring equation ... is only an
approximation to the behavior of a pendulum, which is in fact better
described by the equation ... It is impossible to write down a
solution of this equation.
a. Starting from y(0)=0.1 and ...=0 at t = 0, the solution of the spring equation is y(t) = 0.1 cos(t). Compare this with the solution of the pendulum equation for one period (from t = 0 to t = 2π). Graph the two solutions.
b. Do the same starting from y(0) = 0.2.
c. Do the same starting from y(0) = 0.5.
d. Do the same starting from y(0) = 1.0.
e. Do the same starting from y(0) = 1.5. f.
How long does it take the pendulum to swing all the way back in each
case? Does the period of a pendulum depend on the amplitude?
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2.10.42 Consider the family of functions h ( t ) = cos(3.0t) + 1.5 cos(3.6t) + 2.0 cos(vt) for various values of v ranging from 2.0 to 3.0. Graph them. Why do they look so weird? When do they look least weird?
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2.10.43 Find
the derivatives of the following functions. Where are the critical
points and points of inflection? What happens as more and more cosines
are piled up? Explain this in terms of the updating function ...
a. cos(cos(x)).
b. cos(cos(cos(x))).
c. cos(cos(cos(cos(x)))).
d. cos(cos(cos(cos(cos(x))))).
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