AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions

Transcription

1 AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions Question No. 1 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as 1. Which of the following parametric equations corresponds to the graph? Question #01 (A) { x = t, y = t +, t [,]} (B) { x = t, y = t +, t [,]} (C) { x = t, y = t +, t [ 1,1]} (D) x + = y x = y (E) Try putting a value of t, say t = 1, into the equations. B. Correct! This is the correct answer. It can be found by plotting points. The equations are correct but the range of t is not correct. This is a rectangular equation. (E) Incorrect! This is a rectangular equation. The representation { x = t, y = t +, t [,]} is the correct answer. This can be seen by plotting points: t x y The correct answer is (B).

2 Question No. of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as. Given { x = sin( t), y = cos( t)}, which of the following is the rectangular equation? Question #0 (A) x + y (B) x + y = 1 (C) x y + = 1 x y (D) + = 1 (E) sin( x) = cos( y) This is not an equation. Try using a trigonometric identity. Try using a trigonometric identity. D. Correct! This is the correct answer. It can be found by writing x/ = sin(t) and y/ = cos(t) and then using a trigonometric identity. Try using a trigonometric identity. Given, { x sin( t), y cos( t)} sin ( t) cos ( t) 1 = = we can write sin( t) + =, we have The correct answer is (D). x y + = 1. x y = and cos( t) =. Now, since

3 Question No. of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as. Find dy/dx where { x = t t, y = + t } Question #0 (A) (B) (C) (D) (E) 1 t 1 t 1 t (1 t, ) This is dx/dy, not dy/dx. Recall that dy/dx = (dy/dt)/(dx/dt). Recall that dy/dx = (dy/dt)/(dx/dt). D. Correct! This is the correct answer. It is found by dy/dx = (dy/dt)/(dx/dt). (E) Incorrect! This is an ordered pair. The answer should be a function of t. Given the parametric equations { x = t t, y = + t }, the slope of the tangent line in dy dy. dx dx 1 t dt the xy plane is given by = = The correct answer is (D). dt

4 Question No. 4 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as Question #04 4. Find the equation of the tangent line to { x = 4t + 1, y = t } when = (B) y = 7x + 9 (A) y = x (C) y = x + (D) y = 6 (E) y = 9x t. A. Correct! This is the correct answer. It is found finding the slope of the tangent line dy/dx and then using the point-slope formula. First find the slope of the tangent line dy/dx. Then use the point-slope formula. First find the slope of the tangent line dy/dx. Then use the point-slope formula. First find the slope of the tangent line dy/dx. Then use the point-slope formula. First find the slope of the tangent line dy/dx. Then use the point-slope formula. 1 x t y t when t = Given = + = { 4 1, } dy t t = = = dx 8t 8 8 t =. Next, we find the point in the xy plane: x() = 4() + 1 = 7 1 y() = () = 9 Lastly, we use the point slope formula:, we first find the slope of the tangent line: y y = m( x x ) 1 1 y 9 = ( x 7) 8 y = ( x 7) y = x 8 8 The correct answer is (A).

5 Question No. of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as. Let { x = cos( t), y = sin( t), t [0,1]}. Find the arc length. Question #0 (A) (B) 1 (C) (D) (E) A. Correct! This is the correct answer found by the arc length formula. Substitute into the arc length formula. A trigonometric identity may be helpful. A Feedback on Each Answer Substitute into the arc length formula. A trigonometric identity may be helpful. Substitute into the arc length formula. A trigonometric identity may be helpful. Substitute into the arc length formula. A trigonometric identity may be helpful. Given { x = cos( t), y = sin( t), t [0,1]}, we find x' = sin( t ) and y ' = cos( t ). Plugging these into the arc length formula yields: = + 1 ( x ') ( y') dt ( sin( t)) (cos( t)) dt = 4sin 1 ( 1 t ) + 4cos ( t ) dt = 4 dt = =. The correct answer is (A)

6 Question No. 6 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as 6. Which of the following does not describe a relationship between the rectangular coordinate system and the polar coordinate system. Question #06 (A) x + y = r (B) x = rcos( t) (C) x + y = r y (D) tan( t) = x (E) y = rsin( t) This is a valid relationship between the coordinate systems. This is a valid relationship between the coordinate systems. C. Correct! This is the correct answer. x + y = r is not a valid relationship between the rectangular and polar coordinate systems. This is a valid relationship between the coordinate systems. This is a valid relationship between the coordinate systems. By graphing a point in both the rectangular and polar coordinate systems, the following relationships can be derived: x = rcos( t) y = rsin( t) x + y = r y tan( t) = x The correct answer is (C).

7 Question No. 7 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as 7. Let r = cos( t), t = [0, ]. Find all values of t where the tangent line is horizontal. Question #07 (A) t = 4 (B) t = 0 (C) t = (D) t = 4 (E) t = 6 A. Correct! This is the correct answer, found by finding dy/dx and setting it equal to zero. First find dy/dx and then set equal to zero. A Feedback on Each Answer First find dy/dx and then set equal to zero. This answer is not within the domain of the function. First find dy/dx and then set equal to zero.. If r cos( t) =, then x = rcos( t) = cos ( t) and y = rsin( t) = cos( t)sin( t). We find dy dy cos ( t) sin ( t) =. dx cos( t)sin( t) Setting the numerator equal to zero, we have cos ( t) = sin ( t) which implies that t =. 4 dx to be: The correct answer is (A).

8 Question No. 8 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as 8. Let r = cos( t), t = [0, ]. Find all values of t where the tangent line is vertical. Question #8 (A) t = (B) t = 4 (C) t = (D) t = 0 (E) t = 0, First find dy/dx and then set the denominator equal to zero. First find dy/dx and then set the denominator equal to zero. This is a correct answer but not the only one. This is a correct answer but not the only one. E. Correct! This is the correct answer, found by finding dy/dx and setting the denominator equal to zero. If r cos( t) =, then dy cos ( t) sin ( t) =. dx cos( t)sin( t) x = rcos( t) = cos ( t) and y rsin( t) cos( t)sin( t) Setting the denominator equal to zero, we have cos( t)sin( t)) = 0 sin( t) = 0 t = 0, = =. We find dy dx to be: The correct answer is (E).

9 Question No. 9 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as 9. Find the area of the polar curve r = cos( t), t, (A) Question #09 (B) 1 (C) (D) 1 (E) Use the integral formula for the area of a region bounded by a polar curve. B. Correct! This is the correct answer. It is found by using the polar integral formula for area. Use the integral formula for the area of a region bounded by a polar curve. Use the integral formula for the area of a region bounded by a polar curve. Use the integral formula for the area of a region bounded by a polar curve. Given r = cos( t), t, 6 6 A = = /6 /6 /6 /6 1 r dt 1 (cos( )) t dt 1 t = cos( t )sin( t ) = 1 /6 /6, the area is The correct answer is (B).

10 Question No. 10 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as t 10. Find the arc length of the polar curve r = e, [0,] t. Question #10 (A) (B) e (C) (D) + e (E) None of these Use the arc length formula for a polar function. This cannot be the answer because arc length must always be positive. A Feedback on Each Answer Use the arc length formula for a polar function. D. Correct! This is the correct answer. It is found by using the arc length formula for polar curves. One of the given choices is correct. Please try again. Given curve formula, we have r t, we know that r = e t t and ( r') = e. Using the arc length t = e, [0,] The correct answer is (D). t t r + ( r ') dt = e dt = e = e