some easy calculus 1, questions

\f(1 point) For what values of x in [0, 271'] does the graph of f(x) = x + 2 sin x have a horizontal tangent? List the values of x below. Separate multiple values with commas. x: (1 point) (a) Let f (x) = sin(x). Find f' (x). f' (x) = (b) Let f (x) = sin (x4 ) . Find f' (x). f' (x) =(1 point) The displacement of a particle on a vibrating string is given by the equation s(t) = 15 + %sin(127rt), where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds. v(t) = (1 point) At what point does the normal to y = 3x2 (1 + 2x) at (1,0) intersect the parabola a second time? Answer: Note: You should enter a cartesian coordinate. The normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals i.e. if the slope of the first line is m then the slope of the second line is 1/m (1 point) Find the equation of the tangent line to the curve y = 3x cos x at the point (71, 37r). The equation of this tangent line can be written in the form y = mx + b. Compute m and b. m: b: COS X (1 point) Suppose that f ( ) = -6 and f' (T ) = 8, and let g(x) = f (x) sin x and h(x) = Answer the following questions. f(x) 1. Find g' (7/2). Answer: g' (7/2) = 2. Find h'(7/2). Answer: h' (7t/2) =(1 point) Calculate dP if V2R P = ( R+r) 2 where r is variable, and R and V are constant. dp dr