Q1 (3 points) The graph below shows the position of a particle on the number line over a time period of 10 minutes (time is measured on the horizontal axis, while position is measured on the vertical axis}. Answer the following questions, using appropriate units and briefly explaining your answers. a} What is the displacement of the particle over the entire 10 minutes? b} What is the average velocity of the particle over the entire 10 minutes? c} What is the instantaneous velocity of the particle after 1 minute? d} What is the instantaneous speed of the particle after 6 minutes? e} What is the acceleration of the particle after 2 minutes? f) What is the total distance travelled by the particle over the entire 10 minutes? 02 {2 points) Let x) be a twice differentiable function (the first and second derivatives exist]. Determine the first and second derivatives of g(x) = f(x)e~"' + 0050:) in terms of f(x) and its derivatives. No explanation is required. Q3 (2 points) Let f(x) = cos(x) . Calculate f(2023(x) . Explain your answer.Assume that the radius r of a sphere is expanding at a rate of 20 cm/min. The volume of a sphere is V = gmg and its surface area is 47172. Determine the rate at which the surface area is changing with respect to time when r = 40 cm. (Use symbolic notation and fractions where needed.) = cmzfmin dt Find the equation of the tangent line to the function f(x) = In (x ) at x = 3. (Use symbolic notation and fractions where needed.) y =Find the derivative. (Express numbers in exact form. Use symbolic notation and fractions Where needed.) d dx(6x1n(x) 5x) = | Given, 42 Ki f ( x ) = en ( x 5 ) at X= 3. y = f ( 3 ) = en 3 3 = 5 en 3 , Now, slope of f (x ) at 21 = 3. m = f' ( x ) . 5 x 4 5 X - > m = p' ( x ) = = X -> m = p' ( 3 ) = 5 Equation of tangent is y - y1 = m ( x - x 1 ) Putting , X1 = 3 41 = 5 1 3 mc 54 = 5x + 5 ( ln 3-1 ) = 5% + 0. 493 y = 1. 67 x + 0. 493 3 y = 5 x + 0.49 or / y = 1 . 67 x + 0 . 49 3 4. = 5x + 0.5 4 = 1. 67 x + 0.5 3