Calculus derivative

Question

W(t)={17/2−3/2cos(πt/6) for0≤t≤6

10−1/5(t−6)^2. for6<t≤10

The depth of a river at a certain point is modeled by the function  W defined above, where  W(t) is measured in feet and time  t is measured in hours.

(a)  Find  W′(8). Using correct units, explain the meaning of  W′(8) in the context of the problem.

(b)  The graph of  W is concave down for  3≤t≤3.5. Use the line tangent to the graph of  W at t=3 to show that  W(3.5)≤9.

Answer 1

w'= 3pi/12sin(pit/6) 0<=t<=6 ... w' = -.4(t-6), 6<t<=10 .... w'(3)= 3Pi/12sin(pi/2)= 3pi/12 ...

a) w'(8)=-.4(8-5)=-.8 indicating that the depth is decreasing by .8 feet per hour

b) w'(3) = 3pi/12sin(3pi/6) = 3pi/12 = slope of tan line at t=3. w(t) = 3pi/12t+b . w(3)= 17/2-3/2cos(3pi/6)= 17/2 thus b = 17/2-9pi/12=17/2-3pi/4 ... w(3.5) = 3pi/12*7/2+17/2-3pi/4= 21pi/24-18pi/24+17/2=3pi/24+8.5 < 9 ... QED

Answer 2

See attached video and message me if you have additional questions.