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(TT) = 1. (5 points) 4-3cos(x) O -2-3cos(x) 2-3cos(x) O -4-3cos(x) 2. The slope of the tangent to a curve at any point (x, y) on the curve is . . Find the equation of the curve if the point (2, -3) is on the curve. (5 points) O x2 +y2 = 13 O x2+y2 = 25 O x2 - y2 =-5 O x 2 - y2 = 5 3. The rate of decay in the mass, M, of a radioactive substance is given by the differential equation am = -KM , where k is a positive constant. If the initial mass was 200g, then find the expression for the mass, M, at any time t. (5 points) M = 200In(kt) O M = 2e-kt M = 200 ekt O M = 200 e-kt 4. The temperature of a cup of coffee varies according to Newton's Law of Cooling: "= -k(T-A) , where T is the temperature of the coffee, A is the room temperature, and k is a positive constant. If the coffee cools from 100 C to 80 C in 1 minute at a room temperature of 22 C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. (5 points) O 20 O 24 46 54 5. The differential equation = (5 points) 1. produces a slope field with vertical tangents at y = 3 Il. produces a slope field with vertical tangents at x = -1 Ill. produces a slope field with rows of parallel segments I only O Il only O Ill only I and Ill6 . 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 dx = 0 is a family of curves. These curves are all (5 points) O In O hyperbolas O parabolas O ellipses 8. Estimate the value of x3 dx by using the Trapezoidal Rule with n = 4. (5 points) O 10 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 ((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 O 0.6456 0.6919 O 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, right sed to estimate [ f(x)dx . The estimates were 0.8067, 0.9635, 1.0514, 1.0753 and 1.3439, and t 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. 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? (-15. 12 1) -3. -2.4 -1.6 -4.0 9.2 71.5. -1.0 (5 points) Of is concave up from x = -1.5 to x = 1.5. ) fhas 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). 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 1() 5 4 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. ( f(x) is concave upwards for all x. ( f(x) has a point of inflection at x = 0. ((x) passes through the origin. 7. I is a differentiable function on the interval [0, 1] and g(x) = ((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 1(0 1 2 3 -4 O -16 0 4 O 4 Cannot be determined 8. Use the graph of f() = 21 + 5 on the interval [-1, 4] to write the function F(x). where F(x) - J f (!)dt . (5 points) OF(x) = 2x +5 OF(x) =x2 + 5x -6 OF(x) =x2 + 5x- 1 OF(x) = x2 + 6x 9. The velocity of a particle moving along the x-axis is v(1) = cos(21), with t measured in minutes and v(1) measured in feet per minute. To the nearest foot find the total distance travelled by the particle from t = 0 to t = n minutes. (5 points) 10. Find the range of the function F(x) - [ v36 -+3 dt - (5 points) 1-6.01 O [0. 5] 10, 9ri] 10, 187]1. The figure below shows the graph of f', the derivative of the function f, on the closed interval from x = -2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4. (6, 4) 5, 0) Find the x-value where f attains its absolute maximum value on the closed interval from x = -2 to x = 6. Justify your answer. (10 points) B i U Font Family 2. A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10. What is the position of the car when t = 20? You must show your work answer to 3 decimal places. Include units in your answer. (10 points) B i U Font Family 3. Show that f(x) = x3 and g(x) = 200x3 grow at the same rate. (10 points) B i U Font Family4. 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) 0 0.5 1.2 1.5 2 v(1) 0 4.5 7.8 8.3 9.0 B i U Font Family 5. Oil flows into a tank according to the rate F(t) = - , and at the same time empties out at the rate E(t) = InU" , 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) t+2 3 i U Font Family