Sketch the graphs of the following functions using the 1st and 2nd derivative and without using graphing software. You may check your result using desmos.com or similar. 1. f(x) =x2 - 4x+1 2. f(x) =3x2 - x3+ 1 3. f(x ) = 2+ - 5 x +4 4. f(x) =8x3 - x4 5. f ( x ) = - 1 9- X23m :m b. Written example 1. The volume of a sphere is increasing at a rate of 100 [m3/5. How fast is the radius increasing when the radius is 25 cm? 1. Write the rates of changes. V ,. .. "volume ofa sphere" rate of change is d = w dr ' change of time "radius ofa sphere" rate of change is g = w dr change of time Volume of a sphere is V = gm? 2. What we know and what we are looking for. We are given the rate of change of the volume. = 100 cm3/s di We are asked to nd rate of change of radius for a given radius. r:25 cm dr 7:77 cm/s dt We are dealing with a sphere. We know the volume of a sphere is given by the following formula. 4 V= nr3. 3 3. Find the derivative using the formula and implicit differentiation. How is the sphere volume useful? We can take the derivative with respect to t and then use the known rate of change to nd the unknown rate of change. Here is the derivative using implicit differentiation, [ have highlighted the variable parts below. d 4 3(V 3 rrr3) dV .7=i 2.$ 1 dt 3M3\" dt 4. Use the derivative and the related rates. Use the rate of change of volume to find rate of change of radius. We know dv dt =100 cm3 / s and r=25 cm. 100 cm3 / s = = (3(25 cm)?) . ar Solve for the rate of change of the radius. 100 cm3 / s TT (3(25 cm)2) . TT(3(25 cm)2) T (3(25 cm)2) dr 100 cm3 / s 25 100 125 dt cm / s = 4 TT(625 cm2) AT(625 cm2) 25625 IT -cm/s = 0.0127 cm/sUse the formulas found in the similar notes examples for the following. 1. The volume of a sphere is increasing at a rate of 75 cm3 /5. How fast is the radius increasing when the radius is 10 cm? 2. A ladder 5 ft long is leaning against a wall. The bottom of the ladder slides away from the wall at a rate of 0.5 fr/s. How fast is the ladder sliding down the wall when the bottom of the ladder is 3 ft from the wall? 3. CarA is traveling west at 50 mph and car B is traveling north at 60 mph both heading toward the intersection. At what rate are the cars approaching each other when carA is 0.3 mi and car B is 0.4 mi from the inter