#$&* course MTH 279 6/19 00:04 Question: `q001. Find the first and second derivatives of the following functions:
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q002. Sketch a graph of the function y = 3 sin(4 t + 2). Don't use a graphing calculator, use what you know about graphing. Make your best attempt, and describe both your thinking and your graph. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The function is a periodic sinusoid dependent on t. The altitude is 3 units, the frequency is pi/2 radians (90 degrees), and the phase shift is -1/2 degrees. This means that the function is shifted left -1/2 degrees, it repeats every pi/2 rad (90 degrees), and it reaches a maximum of 3, and a minimum of -3.
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q003. Describe, in terms of A, omega and theta_0, the characteristics of the graph of y = A cos(omega * t + theta_0) + k. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A is the altitude of the sinusoid (which is a cosine in this instance), with angular frequency of omega radians/s, phase of theta_0 radians, and phase shift of theta_0/omega seconds.
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q004. Find the indefinite integral of each of the following: • f(t) = e^(-3 t) • x(t) = 2 sin( 4 pi t + pi/4) • y(t) = 1 / (3 x + 2) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: F(t) = -e^(-3t)/3+C X(t) = -cos(4pi*t+pi/4)/(2pi) + C Y(t) = ln(abs(3x+2))/3 + C confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q005. Find an antiderivative of each of the following, subject to the given conditions: • f(t) = e^(-3 t), subject to the condition that when t = 0 the value of the antiderivative is 2. • x(t) = 2 sin( 4 pi t + pi/4), subject to the condition that when t = 1/8 the value of the antiderivative is 2 pi. • y(t) = 1 / (3 t + 2), subject to the condition that the limiting value of the antiderivative, as t approaches infinity, is -1. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: F(t) = -e^(-3t)/3 + 7/3 X(t) = -cos(4pi*t+ pi/4)/(2pi) + 2pi-sqrt(2)/(4pi) Y(t) = ln(abs(3t+2))/3 + C >>> The lim of ln as t->inf. is inf, so the answer is that this is impossible. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q006. Use partial fractions to express (2 t + 4) / ( (t - 3) ( t + 1) ) in the form A / (t - 3) + B / (t + 1). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (2 t + 4) / ( (t - 3) ( t + 1) ) = A / (t - 3) + B / (t + 1) 2t+4 = A(t+1) + B (t-3) >> 2t+4 = At+A+Bt-3B 2t = At + Bt A = 2 - B 4 = A - 3B >> 4 = 2 - 4B >> B = -1/2 A = 2 + ½ = 5/2 A / (t - 3) + B / (t + 1) = (5/2)/(t-3) - (1/2)/(t+1) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q007. The graph of a function f(x) contains the point (2, 5). So the value of f(2) is 5. At the point (2, 5) the slope of the tangent line to the graph is .5. What is your best estimate, based on only this information, of the value of f(2.4)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Using the tangent line, which can be worked out to be y = 0.5x+4, we can approximate the value of f(2.4) to be 5.2. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q008. The graph of a function g(t) contains the points (3, 4), (3.2, 4.4) and (3.4, 4.5). What is your best estimate of the value of g ' (3), where the ' represents the derivative with respect to t? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (4.5-4.4) = m(3.4-3.2) >> m = 0.5 (4.4-4) = m(3.2-3) >> m = 2 The slope between (3.2, 4.4) and (3.4,4.5) is calculated as 0.5. The slope between (3,4) and (3.2, 4.4) is calculated as 2. By graphing these points with the connecting slopes, we can tell that the graph is increasing at a decreasing rate (at the top of the hill) from left to right. Going back to the point (3,4), we can guess that m = g’(3) > 2. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK "