Math 136 Exam 1 Practice Problems

Transcription

1 Math Exam Practice Problems. Find the surface area of the surface of revolution generated by revolving the curve given by around the x-axis? To solve this we use the equation: In this case this translates to the integral: π x t ln(t), y t, t, β (dx SA π y(t) α dt t (t t ) + dt π ) + ( ) dt dt t (t 8 + t + dt The trick to evaluate this integral is to recognize that the function inside the square root can be written as a complete square. t t (t + t ). Using this and the fact, we can write: SA π π t (t + t ) dt π t + dt ( ) π t + 8πt t(t + t )dt π π ()()π () + π 8π + π. The graph of the polar curve r θ, starting from θ is a outward spiral starting at the origin. Find the length of the portion of this spiral that is inside the circle r that starts at θ. To find the length of this spiral, one must first find where the spiral intersects the circle r. To do this we set θ and find that this is true for θ. Thus the length that we are after can be found by integrating: θ + θ dθ θ θ + dθ θ θ + dθ To compute this integral we can use a u-substitution with u θ +, and du θ. Thus our integral becomes: 8 8 u du u ) (8 8

2 . Find the area inside both curves r + cos(θ) and r + sin(θ). +sin(θ) +cos(θ) Looking at the graph we can identify that these two curves intersect when + cos(θ) + sin(θ) which happens when θ π/ and θ 5π/. (You can solve for these by setting the two equations equal and finding that tan(θ). Also, if we note that the area inside both curves is symmetric with respect to the line corresponding to θ π/, then we can compute the area by integrating from π/ to π/ of just the one curve r + sin(θ) and then doubling it to get the full area. Thus the area we want is: π/ π/ ( + sin(θ) ) dθ π/ π/ π/ π/ 9 + sin(θ) + sin (θ)dθ 9 + sin(θ) + cos(θ)dθ (θ cos(θ) sin(θ)) π/ π/ ( ( ) ) ( ( ) ) π π + π 7.58 If one did not recognize or was unsure that the graph was symmetric, you could instead compute the sum of the following two integrals: π/ π/ The result would be the same number. 5π/ ( + sin(θ) ) dθ + π/ ( + cos(θ) ) dθ

3 . A tank is constructed so that the top is a by meter rectangle. The depth of the tank is meters at the deep end and gradually increases to a depth of at the other end. A figure of the pool is below. What is the force due to hydrostatic pressure on one of the triangular sides of the tank assuming the tank is completely full? meters meters meters Look at the triangular side of the tank as a triangle with the lower left corner at the origin. Then consider a slice of height y at a height y. Either by writing down an equation for the line described by the hypotenuse of the triangle or by using similar triangles then the width of the slice is x y/. So the area of the slice is approximately a rectangle and is given by y y. Thus the force on this slice is approximately ρ g ( y) y y. Note that the depth is given by y. To calculate the entire force we integrate from y to y giving y-axis depth is -y width is Δy y x y(/)x x-axis ρ g y y ρ g ) (y y ρ g ( () / ) 7 Newtons

4 5. The parametric representation of a cycloid is given by x t + sin(t) and y + cos(t), what is the area bounded by the cycloid and the x axis for < t < π? The formula for the area bounded by a curve and the x-axis is given by:. Expanding this we have: β α y(t)x (t)dt π ( + cos(t))( + cos(t))dt π + 8 cos(t) + cos (t)dt π + 8 cos(t) + + cos(t)dt (t + 8 sin(t) + sin(t)) π π. Sketch a graph of each of the following polar graphs. Label at least three points on each graph (other than the origin). In the last two parts your points and your graph will depend on the values of the constants, a, b and c. (a) r cos(θ). (/, Π./) (, ) (-/, Π/)

5 (b) r sin(θ). (, Π/) (,) (c) r a + b cos(θ) where a b >. a, Π a, Π a, 5

6 (d) r c cos(θ) where c is a positive number. (c/, Π/) c, c, Π 7. Explain in complete sentences why the formulas x r cos(θ) and y r sin(θ) are the correct formulas for converting polar to cartesian coordinates. You may include a diagram to accompany your text. If we have a point P, which is given by (x, y) in cartesian coordinates and (r, θ) in polar coordinates then the values of are all parts of a right triangle with the longest side of the triangle (or the hypotenuse) having length r. The horizontal component of the triangle has length x and the vertical component is y. Or x is the side adjacent to the angle θ and y is the side opposite θ. This is illustrated in the diagram below: r y Θ x Since the angles are related in this way, then by definition of cos(θ) it is equal to the opposite side over the hypotenuse, or cos(θ) x/r. Solving for x gives x r cos(θ). Similarly we get y sin(θ). 8. Convert each of the following cartesian points to polar coordinates. (a) (/, /) Here r (/) + ( /) /. We also have that tan(θ) y/x. Thus our reference angle is π/. Since the point is in the fourth quadrant this gives us the correct angle and we can write the point as (/, π/) or as (/, 7π/). (b) (, ) Here r 9 +. We also have tan(θ) y/x. If we compute arctan( / ) we obtain π/. However, this angle is in the wrong quadrant. The angle we need is in the second quadrant and has a reference angle of π/. Thus, θ 5π/. Our point in polar coordinates is thus (, 5π/).

7 9. Find the length of the curve x y 8 + y between the two points ( 8, ) and (, ). This problem is set up to integrate in y so we will use the arc-length formula: d (dx ) (y L + ) y + Expanding what is under the radical we have: Replacing this in the radical gives: c ( y ) y + y + y + y + + ( y y + ) y L y (y + ) y + y ( ) y 8 y ) ( + ( 8 ). Find the equation of the tangent line to the polar curve r cos(θ) at the polar point (/, π/). Give the equation in cartesian coordinates. The formula for the derivative can be computed by first converting to x(θ) cos(θ) cos(θ) and y(θ) cos(θ) sin(θ). We then have: dx dθ dx dθ sin(θ) sin(θ) + cos(θ) cos(θ) sin(θ) cos(θ) cos(θ) sin(θ) ) If I evaluate this at θ π/ then I have sin(π/) sin(π/) + cos(π/) cos(π/) dx sin(π/) cos(π/) cos(π/) sin(π/) ( /) + ( /) ( /) (/) 7 Thus, the slope at this point is given by m 7. The cartesian coordinates at this point are x (/) cos(π/) and y (/) sin(π/) /. The equation of the tangent line can then be found using the point slope formula: y 7 ( x ) or y 7 x + 7 7

8 . True or False, If a parametric curve is given by x(t) and y(t) and x () y () then there is neither a vertical nor a horizontal tangent line when t. Explain your answer. False, If both /dt and dx/dt are then one must look at the limit lim t dx, to determine the slope at that point. Since this is a / type problem you will either have to try to use algebra to simplify or use L Hospitals rule to evaluate this limit. If the limit is then there is a horizontal tangent at t. If the limit is then there is a vertical tangent at this point. If the limit is some other number, then the slope is equal to this number at t.. Given the parametric equations x t + 9t t and y t 8t + t +, find all locations (give the x and y values) when the slope is horizontal. We begin by computing the following: dx dt t + 8t (t + t ) (t + )(t ), which is when t and t. Also, dt t t + t t(t t + ) t(t ), which is when t or t. Since we have that dx/dt and /dt when t, then there is a vertical tangent line or a cusp. Since we have that dx/dt and dx/dt when t then there is a horizontal tangent line here. At t both /dt and dx/dt, thus we must check the limit of /dx to decide what the slope is here. lim t dx lim /dt t dx/dt lim t(t ) t (t + )(t ) lim t(t ), t (t + ) Thus, we have a horizontal tangent at this point. 8

9 . Find the equation of a hyperbola with Foci F (, ) and F (7, ) and vertices V (, ), V (, ). Also give the equations of the asymptotic curves. The center is half way between either the foci or the vertices, so in this case is at (, ). The values a and c are the distances from the center to the vertices and the foci respectively and in this case they are a and c. Thus b c a 9 5. Consequently our equation is: (x ) (y ) 5 The asymptotes are given by y k ± (b/a)(x h) ± 5 (x ).. Decide if the equation x +x+y +y describes a circle, an ellipse, a parabola or a hyperbola. If it describes a circle find the center and radius. If it describes an ellipse find the center foci and vertices. If it describes a parabola find the focus, the vertex and the directrix. If it describes a hyperbola find the foci, vertices and the asymptotes. If we complete the squares from the equation above we have: (x + x + ) + (y + y + ) This simplifies to or Dividing both sides by 5 gives: (x + x + ) + (y + y + ) ( x + ) + (y + ) 5 ( x + ) (y + ) or ( x + 5 ) + (y + ) 5, Thus this is an ellipse. The center of the ellipse is at ( /, ) and since 5 is larger than 5/, then the major axis is parallel to the y-axis and is equal to x /. Here a 5 and b 5/. Thus c 5 5/ 5/. The foci are then ( /, ± 5/) and the vertices are ( /, ± 5). 9

10 Outline of Topics Covered. Chapter 9 (a) Section 9. Length of curves. Be able to compute the length of curves by integrating in either x or y. (b) Section 9. Surface areas of solids of revolution. Be able to compute the surface area of a surface generated by revolving a curve around either the x-axis or the y-axis. (c) Section 9. Applications of integration including hydrostatic force/pressure and center of mass.. Chapter (a) Section. Parametric Equations. Know what a parametric equation is and how to find its graph. (b) Section. Calculus of Parametric Equations. Know how to compute the arc length of a curve given parametrically. Know how to compute the area bounded between a parametric curve and the x-axis. Know how to find the /dx as a function of t for a parametric equation. Be able to find equations of tangent lines as well as points where parametric curves have horizontal or vertical tangent lines. Remember to check the limit of /dx if both /dt and dx/dt equal. (c) Section. Polar Coordinates. Understand how to find the graph of a polar curve. Be able to covert from polar to cartesian coordinates and from cartesian to polar. Given a set of polar coordinates be able to find a equivalent expression where the angle is in a certain range. (d) Section. Calculus with Polar coordinates. Be able to compute areas bounded inside polar curves (or inside one and outside another etc.). Be able to compute the length of polar curves. These problems often involves finding points of intersection of curves. Be able to find the slope of a polar curve by converting to a parametric equation (x(θ) r(θ) cos(θ), y(θ) r(θ) sin(θ)). Be able to find the equation of the tangent line (in cartesian coordinates). Also be able to find where the curve has horizontal and vertical tangents. Be careful to compute the limit at points where both dx/dθ and /dθ are both. (e) Section.5 Conic Sections. Know how to identify if an equation is a parabola, hyperbola or an ellipse (or not a conic section). Be able to find the equation of a parabola, ellipse or hyperbola given important points or equations (foci, vertices, directrix, etc.). Be able to take an equation and put it into standard form to identify the important parts of a conic (this often involves completing the square).