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Questions and Answers
What condition must be met for the function g(x) to increase, based on the Fundamental Theorem of Calculus?
What condition must be met for the function g(x) to increase, based on the Fundamental Theorem of Calculus?
g(x) is increasing if g'(x) = f(x) > 0.
Explain the significance of the rectangle with height f(c) in relation to the integral of f on the interval [a, b].
Explain the significance of the rectangle with height f(c) in relation to the integral of f on the interval [a, b].
The rectangle area f(c)(b-a) equals the integral ∫ab f(x) dx, demonstrating the Mean Value Theorem for Integrals.
Identify the local maximum of g(x) = ∫1x √(t2-1) dt and explain your reasoning.
Identify the local maximum of g(x) = ∫1x √(t2-1) dt and explain your reasoning.
g(x) has a local maximum at x = 0 because g' changes from positive to negative at this point.
Describe how to find the position of the particle at t = π/4 given a(t) = cos(2t) and initial conditions.
Describe how to find the position of the particle at t = π/4 given a(t) = cos(2t) and initial conditions.
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What does it imply if the velocity of the particle is zero at t = 0 and the acceleration is cos(2t)?
What does it imply if the velocity of the particle is zero at t = 0 and the acceleration is cos(2t)?
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What is the expression for the velocity function v(t) derived from the given acceleration function a(t) = cos(2t) with the initial condition v(0) = 0?
What is the expression for the velocity function v(t) derived from the given acceleration function a(t) = cos(2t) with the initial condition v(0) = 0?
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How do you express the definite integral representing the area under the curve f(x) = 2e^(-x^2) from -1 to 1 as a limit of a Riemann Sum?
How do you express the definite integral representing the area under the curve f(x) = 2e^(-x^2) from -1 to 1 as a limit of a Riemann Sum?
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If A denotes the area under the curve f(x) = 2e^(-x^2) between -1 and 1, how is the average value of f over this interval expressed in terms of A?
If A denotes the area under the curve f(x) = 2e^(-x^2) between -1 and 1, how is the average value of f over this interval expressed in terms of A?
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Given the integral ∫12 e-2x dx, what is the final result after evaluation?
Given the integral ∫12 e-2x dx, what is the final result after evaluation?
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What are the values of x at which the absolute maximum and minimum of g(x) = ∫0xf(t) dt occur over the interval [-5, 6]?
What are the values of x at which the absolute maximum and minimum of g(x) = ∫0xf(t) dt occur over the interval [-5, 6]?
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Study Notes
Practice Test - Unit 6: Integration and the Fundamental Theorem of Calculus
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Statement I: The function g(x) = ∫f(t) dt is increasing when the graph of y = f(x) is above the x-axis. This is True. The derivative of g(x) is g'(x) = f(x). If f(x) > 0, g(x) is increasing.
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Statement II: The integral ∫v(t) dt, where v(t) is velocity, gives the particle's change in position on [a, b]. This is True. By definition, velocity is the rate of change of position. Integrating velocity over an interval gives the change in position.
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Statement III: If a function f is continuous and positive on [a, b], there exists a number c ∈ [a, b] such that the area of the rectangle f(c)(b - a) equals ∫abf(x) dx. This is True. This is the Mean Value Theorem for Integrals.
Problem 2: Local Maxima
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Find all values of x for which the function g has a local maximum. g(x) = ∫0x√(t2-1) dt.
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To find local maxima, determine critical points where g'(x)=0.
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Critical points found by setting g'(x) = √(x2-1), which is the integrand, equal to 0 x = -1, x = 0, x = +1.
Problem 3: Acceleration and Position
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The acceleration of a particle moving along a horizontal coordinate is given by a(t) = cos(2t), t ≥ 0.
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At t = 0, the velocity v(0) = and position s(0) = 0.
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Determine the position of the particle at t = π/4.
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Find the velocity equation (v(t)) then use this to solve position. Then deduce if the particle is moving towards or away from the origin.
Problem 4: Area and Definite Integrals
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Function f(x) = 2e-x2 on the interval −1 < x < 1.
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Find a definite integral to represent the area.
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Express the definite integral as the limit of a Riemann sum.
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Express the average value of f over −1 < x < 1 in terms of the area (A).
Problem 5: Evaluate Definite Integrals
- Evaluate several definite integrals using appropriate techniques.
Problem 6: Absolute Maximum and Minimum
- Find the absolute maximum and minimum values of g(x) = ∫0xf(t) dt on the interval [-5, 6]
Problem 7: Derivative and Function
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The derivative of function f is graphed. Determine the value of f(8), given f(0) = 8
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Then calculate the slope of the line tangent to f at x = 8
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Description
Test your understanding of integration and the Fundamental Theorem of Calculus with this quiz. Explore concepts including the relationship between function behavior and integral values, as well as conditions for local maxima in integral functions. Perfect for those studying calculus at an advanced level.
