MATH 31A HOMEWORK 9 (DUE 12/6) PARTS (A) AND (B) SECTION 5.4. f(x) = x + 1 x 2 + 9, F (7) = 0
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1 FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION ) Express the antiderivative F (x) of f(x) satisfying the given initial condition as an integral. f(x) = x + 1 x 2 + 9, F (7) = 28.) Find G (1), where G(x) = x 2 t3 + 3 dt. 32.) Calculate d x 4 dx x t dt ) Let A(x) = x f(t) dt, with f(x) as in Figure 11. (a) Where does A(x) have a local maximum at P? (b) Where does A(x) have a local minimum? (c) Where does A(x) have a local maximum? (d) True or false? A(x) < for all x in the interval shown. 42.) Find a b such that b a (x2 9) dx has minimal value. 44.) Let A(x) = x f(t) dt, and match the property of A(x) with the corresponding property of the graph of f(x). Assume f(x) is a differentiable.
2 FROM ROGAWSKI S CALCULUS (2ND ED.) 48.) Figure 13 shows the graph of f(x) = x sin x. Let F (x) = x t sin t dt. (a) Locate the local max and absolute max of F (x) on [, 3π]. (b) Justify graphically: F (x) has precicely one zero in [π, 2π]. (c) How many zeros does F (x) have in [, 3π]? (d) Find the inflection points of F (x) on [, 3π]. For each one, state whether the concavity changes from up to down or from down to up. SECTION ) A particle moves in a straight line with the given velocity. Find the displacement and distance traveled over the time interval, and draw a motion diagram like Figure 3 (see chapter
3 for example) (with distance and time labels). v(t) = 36 24t + 3t 2, [, 1] 13.) Find the net change in velocity over [1, 4] of an object with a(t) = 8t t 2 m/s ) Show that if acceleration is constant, then the change in velocity is proportional to the length of the time interval. 2.) To model the effects of a carbon tax on CO 2 emissions, policy-makers study the marginal cost of abatement B(x), defined as the cost of increasing CO 2 reduction from x to x + 1 tons (in units of ten thousand tons - Figure 4). Which quantity is represented by the area under the curve over [, 3] in Figure 4? 24.) The heat capacity C(T ) of a substance is the amount of energy in joules required to raise the temperature of 1 g by 1 C at temperature T. (a) Explain why the energy required to raise the temperature from T 1 to T 2 is the area under the graph of C(T ) over [T 1, T 2 ]. (b) How much energy is required to raise the temperature from 5 to 1 C if C(T ) = T? 1.) Show that the function is concave up on the whole real line. AND THE FOLLOWING PROBLEMS: f(x) = x (sin 4 t + 5t) dt 2.) A particle moves along a straight line. Its displacement at time t from an initial point on the line is given by the formula f(t) = 1 2 t2 + 2t sin t for t π. From t = π and afterwards the particle moves with constant velocity, i.e. the velocity it acquires at time t = π. Compute the following:
4 (a) Find its velocity at t = π (b) Find its acceleration at t = 1 2 π. (c) Find its displacement from at time 5 2 π FROM ROGAWSKI S CALCULUS (2ND ED.) (d) Find a time t > π when the particle returns to the initial point, or else prove that it never returns to. PART II SECTION ) Use the change of variables formula to evaluate the definite integral ) Evaluate 2 r 5 4 r 2 dr. 1 4x + 12 (x 2 + 6x + 1) 2 dx SECTION ) Find the area of the shaded region in the figure. 22.) Figure 18 shows the region enclosed by y = x 3 6x and y = 8 3x 2. Match the equations with the curves and compute the area of the region.
5 In 29 and 38, sketch the region enclosed by the curves and compute its area as an integral along the x- or y- axis. 29.) x + y = 4, x y =, y + 3x = 4 38.) x + y = 1, x 1/2 + y 1/2 = 1 48.) Express the area (not signed) of the shaded region in Figure 2 as a sum of three integrals involving f(x) and g(x). SECTION ) Find the volume of the wedge in Figure 2(A) by integrating the area of vertical cross sections. 8.) Let B be the solid whose base is the unit circle x 2 + y 2 = 1 and whose vertical cross sections perpendicular to the x-axis are equilateral triangles. Show that the vertical cross sections have area A(x) = 3(1 x 2 ) and compute the volume of B. 14.) Find the volume of the solid with the given base and cross sections. The base is the region enclosed by y = x 2 and y = 3. The cross sections perpendicular to the y-axis are rectangles of height y 3. 2.) A plane inclined at an angle of 45 passes through a diameter of the base of a cylinder of radius r. Find the volume of the region within the cylinder and below the plane. 22.) Let S be the intersection of two cylinders of radius r whose axes intersect at an angle θ. Find the volume of S as a function of r and θ.
6 FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION ) Find the volume of revolution about the x-axis for the given function and interval. f(x) = 4 x 2, [, 2] In 16 and 18, (a) sketch the region enclosed by the curves, (b) describe the cross section perpendicular to the x-axis located at x, and (c) find the volume of the solid obtained by rotating the region about the x-axis. 16.) y = x 2, y = 2x ) y = 1 x, y = 5 2 x 26.) Let R be the region enclosed by y = x 2 + 2, y = (x 2) 2 and the axes x = and y =. Compute the volume V obtained by rotating R about the x-axis. Hint: Express V as a sum of two integrals. 3.) Find the volume of the solid obtained by rotating region A in Figure 13 about the y-axis. 46.) Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. y = 2 x, y = x about y = 4 56.) The torus (doughnut-shaped solid) in Figure 15 is obtained by rotating the circle (x a) 2 + y 2 = b 2 around the y-axis (assume that a > b). Show that it has volume 2π 2 ab 2. Hint: Evaluate the integral by interpreting it as the area of a circle.
7 AND THE FOLLOWING PROBLEMS: 1.) Find c such that the area between the curves x 4 c and c x 4 is 1. 2.) Compute the volume of the solid formed by rotating the region bounded by y = 4 x 2, y = 2 + x, x = 2, and the y-axis around the line y = 4.
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