D. Correct! This is the correct answer. It is found by dy/dx = (dy/dt)/(dx/dt).

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1 Calculus II - Problem Solving Drill 4: Calculus for Parametric Equations Question No. of 0 Instructions: () 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) t t t ( t, ) A. Incorrect! This is dx/dy, not dy/dx. Recall that dy/dx = (dy/)/(dx/). Recall that dy/dx = (dy/)/(dx/). D. Correct! This is the correct answer. It is found by dy/dx = (dy/)/(dx/). (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 t the xy plane is given by = =

2 Question No. of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as t. Find the slope of the tangent line to { x = e, y = t } when t = 4. Question #0 (A) e 4 (B) 6 8 e (C) 6 4 (D) e 8 e (E) 8 4 A. Incorrect This is just x(4). The slope of the tangent line is given by dy/dx. This is just y(4). The slope of the tangent line is given by dy/dx. First find dy/dx then plug in t = 4. D. Correct! This is the correct answer, found by evaluating dy/dx at t = 4. This is dx/dy, not dy/dx. t The slope of the tangent line to { x = e, y = t } when t = 4 is given by: dy t t 4. dx e e e dy = = = = dx t t 8 t= 4 t= 4

3 Question No. of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as Question #0. Find the equation of the tangent line to = + = (A) y = x (B) y = 7x + 9 (C) y = x (D) y = 6 (E) y = 9x + 7 x t y t when t =. { 4, } 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. Given { x = 4t +, y = t } when t =, we first find the slope of the tangent line: Next, we find the point in the xy plane: Lastly, we use the point slope formula: dy t t = = = dx 8t 8 8 t = x() = 4() + = 7 y() = () = 9 y y = m( x x ) y 9 = ( x 7) 8 y = ( x 7) y = x 8 8..

4 Question No. 4 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 4. Given { x = cos( t), y = sin( t), t [0, π]}, at what value(s) of t will the tangent line be horizontal? Question #04 (A) t = 0 π π (B) t =, π (C) t = (D) t = π (E) t = π A. Incorrect! First find dy/dx and then set it equal to zero. This would be the correct answer if the domain where 0 to *Pi. C. Correct! By finding where dy/dx = 0 in the domain, you obtained the correct answer. This value of t is not in the domain. First find dy/dx and then set it equal to zero. We first find dy : dx dy = cos( t ). dx sin( t) Then we set it equal to zero: cos( t) π = 0 cos( t) = 0 t =. sin( t)

5 Question No. 5 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 5. Given parametric equations { x = x( t), y = y( t), t [ a, b ]}, which of the following represents area in the xy plane? Question #05 (A) dy dx (B) ydx (C) y (D) x (E) ( x ') + ( y') A. Incorrect! This represents the slopes of tangents in the xy plane. B. Correct! This is the formula for area in the xy plane. The differential should be in terms of x. The differential should be in terms of x. This is a variant of the arc length formula. In the xy plane, area for a region defined by parametric equations is given by the same integral as for one variable functions. This integral is ydx. To evaluate this integral, convert everything into terms involving the parameter t.

6 Question No. 6 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 6. Consider the region under the curve { x = ln( t), y = t, t [,5]}. Find the area of this region. Question #06 (A) ln() (B) 4 (C) ln(4) (D) 0 (E) A. Incorrect! Integrate the expression y*dx by first converting everything in terms of t. Integrate the expression y*dx by first converting everything in terms of t. choice Integrate the expression y*dx by first converting everything in terms of t. By looking at the graph, it is clear that the area is not zero. E. Correct! This the correct answer. You integrate the expression y*dx by converting everything in terms of t. Given { x = ln( t), y = t, t [,5]}, recall that the area in the xy plane is ydx. Since x = ln( t ), dx = t 5 5 so that ydx = t = t t t = t = (5 ) = (5 ) =.. When t [, 5], we have

7 Question No. 7 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 7. Let { x =, y = t, t [,]}. Find the arc length. Question #07 (A) 6 (B) (C) (D) (E) None of these A. Correct! This is the correct answer. It can be derived either by the arc length formula or by sketching the graph. Try sketching the graph. Try sketching the graph. Try sketching the graph. One of the given answers is correct, please try again. We can easily sketch the graph of { x =, y = t, t [,]} and calculate the arc length to be 6: Alternatively, we may use the arc length formula: ( + = + = = x ') ( y ') (0) () t = 6.

8 Question No. 8 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 8. Let { x = cos( t), y = sin( t), t [0,]}. Find the arc length. Question #08 (A) (B) (C) π (D) π (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,]}, we find x' = sin( t ) and y ' = cos( t ). Plugging these into the arc length formula yields: = + ( x ') ( y') ( sin( t)) (cos( t)) = 4sin ( t ) + 4cos ( t ) = 4 = =

9 Question No. 9 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 9. Given parametric equations { x = x( t), y = y( t), t [ a, b ]}, if the curve is revolved around the y axis, then the surface area of the resulting surface of revolution is which of the following? Question #09 b (A) π x ( x') + ( y ') a b (B) π y ( x') + ( y') a b (C) ( x ') + ( y') a dy (D) = dx dy dx (E) ydx A. Correct! This is the correct answer. This would be correct if it were revolved around the x axis. A Feedback on Each Answer This is the formula for arc length. This is the formula for slopes of tangents. This is the formula for area in the xy plane. If the curve { x = x( t), y = y( t), t [ a, b ]} is revolved around the y axis, then the b resulting surface area is given by π x ( x') + ( y '). a

10 Question No. 0 of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as 0. Suppose that the curve { x = t, y = t, t [0,]} is revolved about the x axis. Find the surface area of the resulting surface of revolution. Question #0 (A) π 5 (B) 9 π (C) (D) π (E) 8π 5 A. Incorrect! Use the formula for the surface area when revolving about the x axis. Use the formula for the surface area when revolving about the x axis. Use the formula for the surface area when revolving about the x axis. Use the formula for the surface area when revolving about the x axis. E. Correct! This is the correct answer found by using the formula for the surface area when revolving about the x axis. Given the curve { x = t, y = t, t [0,]}, the surface area of the surface of revolution that results from revolving the curve about the x axis is given by: πy ( x') + ( y') = π( t) + 4 = 4πt 5 = πt 5 = 8π 5

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

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