L(t) (ft^3 /min)^47986520
(a) Estimate L ′(15).
(b) Explain in this context what your answer to part a means.
(c) The falling grain forms a conical pile that the worker estimates to be 5 times as far across as it
is deep. The pile was 3 feet deep when the repairs had been half completed. How fast was the
depth increasing then?
(d) Estimate the total amount of grain that leaked out while the repairs were underway.
- Let f be the function satisfying the differential equation and passing through (0, −1).
(a) Sketch the slope field for this differential equation at the points shown.
(b) Use Euler’s method with a step size of 0.5 to estimate f (1).
(c) Solve the differential equation, expressing f as a function of x.
- Let C represent the arc of the curve determined by P(t) = (9 − t^2 , 2 t ) between its y-intercepts.
Let R represent the region bounded by C and the y-axis. Set up, but do not evaluate, an integral
in terms of a single variable for:
(a) the area of R;
(b) the length of C;
(c) the volume of the solid generated when R is rotated around the y-axis.
- The graph of function f consists of the semicircle and line segment shown in the figure. Define
the area function
