please solve all the questions in the images! Thank you!

1. Find the particular solution to y* = 3sin(x) given the general solution is y = C - 3cos(x) and the initial condition y() = 1. (5 points) O 4-3cos(x) Q 2-3cos(x) O 2-3cos(x) O 4-3cos(x) 2.The slope of the tangent toa curve atany point (x,y)on the curve s X . Find the equation of the curve f the point (2.3)is on the curve. (5 pints) O erp=m 3. The rate of decay in the mass, M, of a radioactive substance i given by the differential equation % - , where ks a positive constant.If the nital mass was 200g, then find the expression for the mass, M, at any timet. (5 points) O M=200in(kt) O m=2ek O M=200ekt O M=200ekt 4. The temperature of a cup of coffee varies according to Newtoris Law of Cooling: &T = (T -), where T i the temperature of the coffee, A s the room temperature, and k i a positive constant.If the coffee cools ffom 100C to 80C in 1 minute at a room temperature of 22C, find the temperature, 10 the nearest degree Celsius of the coffee after 4 minutes. (5 points) @ _x 5 The differentil equation & - X1 5 points) 1. produces a slope field with vertical tangents at 11 produces a slope field with vertical tangents at x =-1 11l produces a slope field with rows of parallel segments Tonly I only Il only Tand it O O O O 6. Which of the following differential equations is consistent with the following slope field? ( 5 points ) o : 7. The general solution of the differential equation dy - 0.2x dix = 0 is a family of curves. These curves are all (5 points) O lines O hyperbolas O parabolas ellipses 8. Estimate the value of x3 dx by using the Trapezoidal Rule with n = 4. (5 points) O 1 O 14 0 6 6.5 9. The table below gives selected values for the function f(x). With 5 rectangles, using the left side of each rectangle to evaluate the height of each rectangle, estimate the value of [f(x)dx _(5 points) * 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 1(x) 1 0.909 0.833 0.769 0.714 0.667 0.625 0.588 0.556 0.526 0.500 O 0.7456 0.6456 0.6919 0.6932 10. Given f(x) > 0 with f "(x) > 0, and f "(x) > 0 for all x in the interval [0. 3] with f(0) = 0.1 and f(3) = 1, the left, rig tions were used to estimate [f(x)dx . The estimates were 0.8067, 0.9635, 1.0514, 1.0753 and 1.3439, and the were used in each case. Match the rule to its estimate. (5 points) 1. midpoint a. 0.8067 2. actual area b. 1.3439 3. trapezoidal C. 0.9635 4. right endpoint d. 1.0753 5. left endpoint e. 1.05141. Write and then solve the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to the square root of y." (10 points) B i U Font Family BcK (XB) 2. Solve the differential equation S= fory = f(x) with the initial condition y(0) = 2. (10 points) B / U Font Family 3. a. Solve the differential equation y' = 6x /1-y . Explain why the initial value problem y' = 6x,/1-y? with y(0) = 2 does not have a solution. (10 points) B i U Font Family 4. The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of ]f(x)dx . Give 3 decimal places for your answer. (10 points) 1 1 1.1 1.2 1.5 1.7 1.9 2.0 1(x) 1 3 4 6 7 8 10 B i U Font Family 5. Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for [x?dix . (10 points) B i U Font Family1. The graph of f'(x) is continuous and increasing with an x-intercept at x = 4. Which of the following statements must be true? (5 points) The graph of f is always concave down. The graph of f is always increasing. ) The graph of f has a relative minimum at x = 4. The graph of f has an inflection point at x = 4. 2. Below is the graph of f'(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1, and x = 2 and a relative maximum at x = -1.5 and a relative minimum at x = 1.5. Which of the following statement is false? -3.1 -2.4 -1.6 -0.0 0.5 1.6 /2.4 3,2 71.5. 1.0 (5 points) Of is concave up from x = -1.5 to x = 1.5. )f has an inflection point at x = 1.5. Of has a relative minimum at x = 2. All of these are false. 3. The graph of y = f'(x), the derivative of f(x), is shown below. List the intervals where the graph of f is increasing. (5 points) O (-4,-2) U (2, 4) O (-2,2) O (-4,0) O (0,4) 4. Which of the following functions grows the fastest as x grows without bound? (5 points) O ((x ) = ex O g(x) = ecosx They all grow at the same rate. 5. Compare the growth rate of the functions f(x) = logs(x) and g(x) = 4%. (5 points) Of(x) grows faster than g(x). O g(x) grows faster than f(x). Of(x) and g(x) grow at the same rate. It cannot be determined.6. f is a function that is differentiable for all reals. The value of f'(x) is given for several values of x in the table below. -8 -3 0 3 8 f ' ( x) 5 4 0 - 2 - 4 If f '(x) is always decreasing, which statement about f(x) must be true? (5 points) Of(x) has a relative maximum at x = 0. Of(x) is concave upwards for all x. f(x) has a point of inflection at x = 0. f(x) passes through the origin. 7. f is a differentiable function on the interval [0, 1] and g(x) = f(4x). The table below gives values of f '(x). What is the value of g'(0.1)? (5 points) 0.1 0.2 0.3 0.4 0.5 f' (x ) 1 2 3 - 4 5 O -16 O -4 O 4 O Cannot be determined 8. Use the graph of f(t) = 2t + 5 on the interval [-1, 4] to write the function F(x), where F(x) - ff (t) at . (5 points) O F (x) = 2x +5 OF(x) =x2 + 5x - 6 OF(x) = *2 +5x- 1 OF(x) = x2+6x 9. The velocity of a particle moving along the x-axis is v(t) = cos(2t), with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance travelled by the particle from t = 0 to t = it minutes. (5 points) 10. Find the range of the function F(x) = [ v36 -tz dt . (5 points) O [-6, 0] [0, 6] [o, 9rt] [0, 18r]1 . The figure below shows the graph ies at x = 2 and x = 4. Find the x-value where f attains its a Justify your answer. (10 points) B / U Font Family "AAA OF . B . . 0. 1 8 2. A car travels along a str leration in ft/sec? is given by the linear graph below for the time interval [0, 301. Att = 0, the velocity of the car is o and its position is 10. what is the posi es. Include units in your answer. (10 points) B / U Font Family " AA- A OF - B . . 9/ 20.1:8 3. Show that f(x) = x3 and g(x) = 200x3 grow at the same rate. (10 points) B / U Font Family " AAA OF . . . 0 0 . . :2 4. A radar gun was used to record the speed of a runner (in meters per second) during selected times in the first 2 seconds of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the runner covered during those 2 seconds. Give a 2 decimal place answer and include units. (10 points) 1 0 0.5 1.2 1.5 2 v(t) 0 4.5 7.8 8.3 9.0 B i U Font Family "AAA OF . B . . 0. . VB 5. Oil flows into a tank according to the rate F() = 5+1 . and at the same time out at the rate Er) = In(7, with both F(t) and E(t) measured in gallons per minute. How much oil, to the nearest gallon, is in the tank at time t = 12 minutes. You must show your setup but can use your calculator for all evaluations. (10 points) B i U Font Family - AA- A D F . B . E . 94 0