Math 261 Final exam practice Sketch the graph of the given function that satisfies all of the given conditions. lim f ( x)  1, lim f ( x)  1, lim f ( x)  0, lim f ( x)  1, f (2)  1, f (0) is undefined x 0 x 0 x 2 x 2 Use the Intermediate Value Theorem to show that there is a positive number c such that c 2  2 . Prove that the equation has at least one real root. sin x  2  x Using the definition of limit, find the slope of the tangent line to the curve y 2 at x  a x3 . x = a. How many tangent lines to the curve y  x /( x  1) pass through the point (1,2)? At which points do these tangent lines touch the curve? A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a ice-cream cone, as shown in the figure. If A(θ) is the area of the semicircle and B(θ) is the area of the triangle, find the following. If 1  f ( x)  x 2  f  x   0 and f(1)=2, find f (1) . 3 Find the derivative of the given function using definition of derivative. State the domain of the function and the domain of its derivative. g ( x)  1  2 x If g ( x)  x sin g ( x)  x 2 and g (1)  0, find g (1) . Find the equation of both tangent lines to the ellipse x 2  4 y 2  36 that pass through the point 12,3 Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is  / 6. . Use the differential (or, equivalently, a linear approximation) to estimate the given number. cos31.5o 99.8 Find the derivatives of the functions. y  cot 2 (sin  ) y  cos(sin 2 x) Find dy / dx by implicit differentiation. y tan( x  y )  1  x2 y sin( x2 )  x sin( y 2 ) Find a formula for D42 x sin x y 2 Graph y  x( x  2) . Find the point on the hyperbola x2 x2  9 xy  8 y x 1 x2 that is closest to the point (3. 0). Find the area of the largest rectangle that can be inscribed in the ellipse: Show that the equation 2 x  1  sin x  0 has exactly x2 y 2   1. 4 9 one real root. Find abs. max and abs. min of the function in the given domain: f ( x)  x  2cos x,    x   A particle is moving with the given data. Find the position of the particle: Find the Riemann sum for f(x) = 2x + 1, 1 x 3, with four terms, taking the sample points to be right endpoints. (Give exact answer. Do not round.) Let g(x) be the following where f is the function whose graph is shown below it. Evaluate g(3). On what interval is g increasing? Evaluate the integral. Where does g have a maximum value? Evaluate the definite integral. The velocity function (in meters per second) is given below for a particle moving along a line. (a)Find the displacement of the particle during the given time interval. (b) Find the distance traveled by the particle during the given time interval. Evaluate the integral. Evaluate the definite integral, if it exists. (Enter NONE if the integral does not exist.) Evaluate the following by making a substitution and interpreting the resulting integral in terms of an area. Suppose f is continuous and the following is true. Find the following. Evaluate the indefinite integral. Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x + y2 = 2 , x + y = 0 y = x4 - x2 , y = 1 - x2 Find the number a such that the line x = a bisects the area under the curve below. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or "washer." x = y - y2 , x = 0; about the y-axis y = x 2/3 , x = 1, y = 0; about the y-axis Refer to the figure and find the volume generated by rotating the given region about the specified line. R1 about OC R1 about BC R2 about OC R2 about BC R3 about OC R3 about BC Find the volume of the described solid S. Use only the variables h, R, and r in your answer. A frustum of a right circular cone with height h, lower base radius R, and top radius r A frustum of a pyramid with square base of side b, square top of side a, and height h A pyramid with height h and base an equilateral triangle with side a (a tetrahedron) Consider the following. (a) Sketch the region and a typical shell. (b) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. (Give your answer correct to 2 decimal places.) Consider the following. Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. (Give your answer correct to 2 decimal places.) Consider the following. (a) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. (Give your answer correct to 2 decimal places.) Consider the following. (a) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the x-axis. (b) about the y-axis. Consider the following. (a) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the x-axis. Consider the following. (a) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method: The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.