#$&* course Mth 277 June 21 round 7:00 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: Solving x = 5 t for t we obtain t = x / 5. Substituting this into the second expression we get y = 3 ( x / 5 ) ^ 2 - 6 which simplifies to give us y = 3/25 x^2 - 6. Note that this is a function which can be graphed in the xy plane, forming a parabola. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q002. Suppose x = 5 t^2 and y = 3 t^4 - 6. Solve x = 5 t^2 for t, then plug this expression in for t in the second equation. Simplify the result. What is your resulting formula for y in terms of x? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = 5t^2 y = 3t^4 - 6 x/5 = t^2 +- sqrt(x)/5 = t y = 3t^4 - 6 y = 3(sqrt(x)/5)^4 - 6 = 3x^2/25 - 6 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: From the first equation we obtain t = +- sqrt( x / 5 ). Plugging this into the second equation, and noting that wither t = + sqrt(x/5) or - sqrt(x/5) the value of t^4 is x^2 / 25, we get y = 3 ( sqrt(x/5) )^4 - 6 = 3 x^2 / 25 - 6. This is the same function we obtained in the first question. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q003. Suppose x = 5 e^t and y = 3 e^(2t) - 6. Solve x = 5 e^t for t, then plug this expression in for t in the second equation. Simplify the result. What is your resulting formula for y in terms of x? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = 5e^t y = 3e^(2t) - 6 x/5 = e^2 y = 3e^(2t) - 6 ln(x/5) = ln(e^t) y = 3e^(2*ln(x/5)) - 6 ln(x/5) = t confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: x = 5 e^t is solved by taking the natural log of both sides, obtaining first ln(x) = ln(5) + ln(e^t). Since the natural log and exponential functions are inverse function, ln(e^t) is just t so we obtain ln(x) = ln(5) + t so that t = ln(x) - ln(5) = ln(x/5). Substituting this into the second equation we get y = 3 e^(4 ln(x/5) ) - 6 = 3 ( e^(ln(x/5) )^2 - 6 = 3 * (x/5)^2 - 6, or y = 3 x^2 / 25 - 6. Note that this is the same function obtained in each of the first three questions. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q004. Let `w = 2 `i - 4 `j + 3 `k, `v = 5 `i + 4 `j - 2 `k. Let `r(t) = `w + t * `v. To orient yourself to these vectors you should: Sketch the `w vector, with its initial point at the origin. Then from the tip, or terminal point, of the `w vector, which is located at the point (2, -4, 3), sketch the `v vector. Now answer the following questions: Let P_0 be the terminal point of the `r vector when t = 0, and let P_2 be the terminal point when t = 2. What is the vector from P_0 to P_2, and what is the unit vector in this direction? What is the unit vector in the direction of `v? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: w = 2i -4j + 3k v = 5i + 4j - 2k let r(t) = w + t*v r(t) = (2 + 5t)i + (-4 + 4t)j + (3 - 2t)k t = 0 r(0) = 2i - 4j + 3k t = 2 r(2) = 12i + 4j - k from P_0 to P_2 r(2) - r(0) = 10i + 8j -4k unit vector in this direction is r(2) - r(0)/ mag. r(2) - r(0) = 10i + 8j - 4k/ sqrt( (12)^2 + (4)^2 + (-1)^2 - (2)^2 + (-4)^2 + (3)^2 ) = 10i + 8j - 4k / sqrt(180) = .75i + .60j - .30k unit vector in direction of v v/ mag.(v) = 5i + 4j -2k / sqrt( (5)^2 + (4)^2 + (-2)^2 ) = 5i + 4j -2k / sqrt(45) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `w = 2 `i - 4 `j + 3 `k, `v = 5 `i + 4 `j - 2 `k. r(t) = w + t v = (2 + 5 t) i + (-4 + 4 t) j + (3 - 2 t) k so r(0) = 2 i - 4 j + 3 k and r(2) = 12 i + 4 j - k. From P_0 to P_2 the vector is r(2) - r(0) = 10 i + 8 j - 4 k. The unit vector in this direction is (r(2) - r(0)) / || r(2) - r(0) || = (10 i + 8 j - 4 k) / sqrt(180) = .77 i + .6 j - .3 k, very approximately. The unit vector in the direction of v is v / || v|| = (5 `i + 4 `j - 2 `k) / sqrt(45) . This is identical to our previous result (10 i + 8 j - 4 k) / sqrt(180) (to see this just factor sqrt(4) out of the denominator and simplify). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q005. How does the solution to the preceding problem support the contention that all of the points 'traced out' by the tip of the `r(t) vector lie along a single straight line? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It is parallel to the v vector from the vector t = point to the t =2 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Thus the vector from the t = 0 point to the t = 2 point is parallel to the v vector. There's nothing special about the t = 2 point, so we conjecture a similar result would apply to the t = 0 point and any other point on the graph. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q006. What are the x, y and z coordinates of the tip of the `r(t) vector? Write in the form x = ..., y = ..., z = ... where you fill in the expressions for ... . Solve the resulting equation for t in terms of x. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: P_0 = (2, -4, 3) v = 5i + 4j - 2k x = 2 + 5t y = -4 + 4t z = 3- 2t solve for t x = 2 + 5t y = -4 + 4t z = 3 - 2t x - 2 = 5t y + 4 = 4t z - 3 = -2t x - 2 /5 = t y + 4 / 4 = t - (z - 3) /2 = t (x - 2) /5 = (y + 4) /4 = - (z - 3) /2 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: r(t) = w + t v = (2 + 5 t) i + (-4 + 4 t) j + (3 - 2 t) k The x, y and z coordinates of the tip of the vector will therefore be x = 2 + 5 t y = -4 + 4 t z = 3 - 2 t Solving the equations for t we get t = (x - 2) / 5 t = (y + 4) / 4 t = -(z - 3) / 2 so we could write (x - 2) / 5 = (y + 4) / 4 = - (z - 3) / 2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q009. Suppose (x - 4) / 3 = (y + 2) / 6 = (z - 2) / 9. What values of x, y and z make these three expressions all equal to zero? If it is known that x = 7, what values of y and z make the other two equations true? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ( x - 4 )/3 = (y + 2) / 6 = (z - 2) / 9 x = 4 y = -2 z = 2 (4, -2, 2) satifies these equations ( x - 4 )/3 = (y + 2) / 6 = (z - 2) / 9 x = 7 y = 4 z = 11 (7 -4) / 3 = 1 (4 + 2) /6 = 1 (11 - 2)/9 = 1 (7,4,11) is also on the line confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: (x - 4) / 3 = 0 if x = 4 (y + 2) / 6 = if y = -2 (z - 2) / 9 = 0 if z = 2. So the point (4, -2, 2) satisfies these equations. If x = 7 then (x - 4) / 3 = (7 - 4) / 3 = 1. It would then follow that (y + 2) / 6 and (z - 2) / 9 , both being equal to (x - 4) / 3, will also be equal to 1. So we have (y+2) / 6 = 1, with solution y = 4 (z - 2) / 9 = 1, with solution z = 11. Thus the point (7, 4, 11) is also on the line. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q010. Your solution to the preceding gives you the coordinates of two points. If `w is the vector from the origin to the first of these points, and `v the vector from the first point to the second, express `w and `v in terms of their `i, `j and `k components. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: w = 4i - 2j + 2k v = (7-4)i + (4-(-2))j + (11-2)k v = 3i + 6j +9k confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: From the origin to the first point (4, -2, 2) the vectors is w = 4 i - 2 j + 2 k. From the first point to the second (7, 4, 11) the vector is v = (7 - 4) i + (4 - (-2) ) j + (11 - 2) k = 3 i + 6 j + 9 k. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q011. If `w = x0 `i + y0 `j + z0 `k and `v = a `i + b `j + c `k, then what, in terms of x0, y0, z0, t, a, b and c, is the expression for the vector `r(t) = `w + t `v? What are the expressions for the x, y and z components of this vector? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: w = x_0i + y_0j + z_0k v = ai + bj +ck r(t) = w + t*v r(t) = (x_0 + at)i + (y_0 + bt)j + (z_0 + ct)k P_0 (x_0, y_0, z_0) v = ai +bj +ck x = x_0 + at y = y_0 + bt z= z_0 + ct confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `r(t) = w + t v = (x0 + a t) i + (y0 + b t) j + (z0 + c t) k so that x = x0 + a t y = y0 + b t z = z0 + c t. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q012. What do you get when you solve the three expressions each for t? What do you get when you set the three resulting expressions equal? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = x_0 + at y = y_0 + bt z = z_0 + ct x - x_0 = at y - y_0 = bt z - z_0 = ct x - x_0 / a = t y - y_0 / b = t z - z_0 / c = t (x - x_0) / a = (y - y_0) / b = (z - z_0) / c confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: From x = x0 + a t y = y0 + b t z = z0 + c t. we get t = (x - x0) / a t = (y - y0) / b t = (z - z0) / c All three expressions are equal to t so they are all equal to one another. Thus (x - x0) / a = (y - y0) / b = (z - z0) / c. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q013. Identify the `w and `v vectors for which the equations (x + 3) / 4 = (y - 2) / 5 = (z + 1) / 3 represent the line `w + s `v. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ( x + 3) / 4 = ( y - 2) / 5 = (z + 1) / 3 x_0 = 3 y_0 = -2 z_0 = 1 a = 4 b = 5 c = 3 (3, -2, 1) in the direction of vector 4i + 5j + 3k confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: As we have seen the general vector r(t) = w + t v has x, y and z coordinates satisfying (x - x0) / a = (y - y0) / b = (z - z0) / c. These equations do not depend on the choice of parameter. The parameter in the expression r(t) = w + t v is t, but the parameter could as well have been s. Had our original expession been r(s) = w + s v we would still have obtained the equations (x - x0) / a = (y - y0) / b = (z - z0) / c. If the equations are (x + 3) / 4 = (y - 2) / 5 = (z + 1) / 3 then we can identify x0 = 3, y0 = -2, z0 = 1, a = 4, b = 5 and c = 3. The meaning is that our equations describe a line through (3, -2, 1) in the direction of the vector 4 i - 2 j + k. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q014. Suppose one straight line is represented by parametric equations x(t) = 3 - 5 t, y(t) = 2 + 4 t and z(t) = -6 - 7 t, while another is represented by the equations x(s) = 2 - 3 s, y(s) = 3 + 6 s and z(s) = -3 - 2 s. If the two lines intersect, it means that for some value of t and some value of x, x(t) = x(s), while for the same values y(t) = y(s) and z(t) = z(s). Show that if t = 1 and s = 2, it isn't so. Express the three given conditions for equality as three simultaneous equations. Do you expect the three equations to have a simultaneous solution? Why or why not? If t = 1 and s = 2: YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x(t) = 3 - 5t x(s) = 2 - 3s y(t) = 2 + 4t y(s) = 3 + 6s z(t) = -6 - 7t z(s) = -3 - 2s t = 1 s = 2 x(1) = 3 - 5(1) x(2) = 2 - 3(2) = -2 = -4 y(1) = 2 + 4(1) y(2) = 3 + 6(2) = 6 = 15 z(1) = -6 - 7(1) z(2) = -3 -2(2) = -13 = -7 = (-2,6,-13) = (-4,15,-7) if we have three equations and three unknows we expect to find a solution. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: our first point is (3 - 5 * 1, 2 + 4 * 1, -6 - 7 * 1) = (-2, 6, -13) and our second point is (2 - 3 * 2, 3 + 6 * 2, -3 - 2 * 2) = (-4, 15, -7). These points are clearly different. The conditions x(t) = x(s), y(t) = y(s), z(t) = z(s) are written out as 3 - 5 t = 2 - 3 s 2 + 4 t = 3 + 6 s -6 - 7 t = -3 - 2 s If we have three equations in three unknowns, we expect to be able to find a solution. If we have three equations in only two unknowns, a solution is not assured. If the numbers in the equations are selected at random, there is a very small chance that there is a solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q015. If the three simultaneous equations in the preceding problem have a solution, find it. If they don't, prove that they don't. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!