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42. For constants a, b, n, R, Van der Waal's equation relates the pressure, P, to the volume, V, of a fixed quantity of a gas at constant temperature T: (P+ n ( V - nb) = nRT. Find the rate of change of volume with pressure, dV/dP.6. Find the tangent line approximation forX] = X sin X near X = int/2. 7. Show that 1 X/Z is the tangent line approximation to 1/ V 1 + a: near X = 0. 12. a. Find the best linear approximation, L(x), to f(x) = e* near x = 0. b. What is the sign of the error, E(x) = f(x) - L(x) for x near 0? c. Find the true value of the function at x = 1. What is the error? (Give decimal answers.) Illustrate with a graph. d. Before doing any calculations, explain which you expect to be larger, E(0.1) or E(1), and why. e. Find E(0.1).18. a. Show that 1+kX is the local linearization of [1 + XY' near x 2 0. b. Someone claims that the square root of 1.1 is about 1.05. Without using a calculator, do you think that this estimate is about right? c. Is the actual number above or below 1.05? In Problems 20-21, the equation has a solution near x 0. By replacing the left side of the equation by its linearization, find an approximate value for the solution.22. a. Find the tangent line approximation near x = 0 to f(x) = 1/(1 - x). b. Use it to approximate 1/0.99.38. Writing g for the acceleration due to gravity, the period, T, of a pendulum of length I is given by T = 27 g a. Show that if the length of the pendulum changes by Al, the change in the period, AT, is given by T AT ~ AL. 21 b. If the length of the pendulum increases by 2%, by what percent does the period change?Assume the equation x y + 2xy = 4 defines y implicitly as a function of x. Use implicit dy differentiation to find the derivative in terms of both a and y. dx O a . dy 2xy3 - 2y dx 3x2y2 + 2x Ob. dy - xy3 dx Oc. dy - -xy3 dx Od. dy 2xy3 + 2y I dx 3x2y2 + 2xAssume the equation cos(xy) = xy defines y implicitly as a function of x. Use implicit differentiation to find the derivative dy in terms of both x and y. dx O a. dy x + y' sin(xy) dx xy + sin(xy) Ob. dy y sin(xy) +y2 = dx x sin(xy) + 2xy O c. dy ysin(xy) +2 dx x sin(xy) + 2xy O d. dy x + y2 sin(xy) dx xy + sin(xy)Assume the equation 3:3 + y3 : 1 defines y implicitly as a function of LC. Note that the point (x, y) : (1, T\") is on the curve defined by this equation (check it and you'll see that it works). Use implicit differentiation to find the slope (derivative) of this curve at this point. _ i _22[3 dd? (z,y)=(1,2\"3) O b. g : _22/3 dd? (z,y)=(1,21/3) O C E _ 22/3 dd? {z,y}:(L2\"3) Q d. : 72/3 6193' {z,y)=(71,2\"3) \fFind the most general antiderivative of That is, find the indefinite integral dx. 1 + x2 1 + x2 O a. dx = In(1 + x2) + C 1 + x2 O b. 1 dx = sin (ac) + C 1 + x2 O C. 1 2x dax = 1+ x2 ( 1 + 2 2)2 + C O d. 1 dx = tan (x) + C 1 + x2Let f($) : :33 + 4x 2. Note that f is a onetoone function {its graph passes the horizontal line test}, so it has an inverse function f1(:B). Also note that f(2) : 23 + 4-2 2 : 8 + 8 2 : 14 so that f1(14) : 2. Use this information to find (f'1)'(14). Find the most general antiderivative of sin (@). That is, find the indefinite integral / sin(x) dx. O a. sin(x) dx = cos(x) + C O b. sin(x) dx = sin(x) + C O C . sin(x) dx = - sin(x) + C O d. sin(x) dx = - cos(x) + CUse your calculator to help you find the error for approximating 111(1.2) by using the tangent line approximation (local linearization} to at) : 111(a:) near 3: : 1. O a. 0.01768 0 b. 0.01768 0 c. 0.01432 0 cl. 0.01432 Consider the following function of two variables: z = f(x, y) = sin(2x + 3y). The partial derivative of this function with az respect to y is: = fy(x, y) = 3 cos(2x + 3y). Note that f 17, 0 = sin = 0.5 and that ay fu (12 0) = 3c05() 3 ~2.59808. 2 Use this information, as well as the multivariable linear approximation when only y changes, f (x, y) ~ f (a, b) + fy (a, b) (y - b) with a = 12 and b = 0, to estimate f ( 12 0.1 ) . O a. f ( 7, 0.1) ~ 0.79 O b. f ( 7, 0.1) ~ 0.77 O c. f ( 7, 0.1 ~ 0.76 O d. f ( 19, 0.1 ~ 0.78Find the tangent line approximation (local linearization) to f(x ) = V1 + x near x = 0. O a. x for * ~ 0 O b. Vita~l- - * for a ~ 0 3 Oc. Vita~ 1 - 3x for x ~ 0 O d. Vita~ 1 + 3x for a ~ 0Find the tangent line approximation (local linearization) to f(a ) = near x = 2. O a. ~ + - x for x ~ 2 4 O b. ~ (x - 2) for x ~ 2 N H N H N H N H O C. for ac ~ 2 O d. + (x - 2) for x ~ 2Assume the equation cos(xy) = xy defines y implicitly as a function of x. Use implicit differentiation to find the dy derivative in terms of both x and y. dx O a . dy y sin(xy) +y2 dx x sin(xy) + 2xy Ob. dy y sin(xy) +y2 dx x sin(xy) + 2xy O c . dy x + y' sin(xy) dx xy + sin(xy) Od. dy x + y2 sin(xy) dx xy + sin(xy)\fFind the derivative of f($) : 111(ta11_1(:1:)) (which is the same function as f(:1:) : 1n(arctan(:1:))). O a 1 . flit? : ) tan1(m) O b. 1 fr 313 : ( ) tan1(m)(1+:r:2) O c. , :1: : 1 f t ) mt1+(1n(:s))2) O . am2 ;17 5602 :1: d f,($):_t () () tanWm)