Review questions for MIDTERM 3.
Use the given graph of to answer the following questions about f:
(a) On what interval(s) is decreasing?
and
(b) List the x- coordinates of all critical points of (in ascending order).
(c) List the x-coordinates of all critical points of that correspond to local
maxima?
(d) List the x-coordinates of all critical points of that correspond to neither local
maxima nor local minima?
(e) On what intervals is concave up?
and
(f) List the x-coordinates of all inflection points of (in ascending order).
(b) Write the x-coordinates of all local maxima of (or write NONE).
(c) Write the x-coordinates of all local minima of (or write NONE).
(d)Find the interval(s) on which is increasing.
(e) Find the interval(s) on which is concave down.
(f) Write the x-coordinates of all inflection points (or write NONE), in ascending
order.
(a) Domain of ,
(b) f is continuous on its domain and differentiable all all points in the domain except at
(c) ,
(d) , , ,
(e) on , and on ,
(f) on ,
(g) on and on ,
(h) on and on
Once you’ve finished, select ‘Done’, and compare your answer with the one shown
(a) List all interval(s) on which is increasing.
(b) List x-coordinates of all points where has a local maximum or write DNE.
(c) List x-coordinates of all points where has a local minimum or write
DNE.
(d) Find .
(e) List all interval(s) on which is concave up.
(f) List x-coordinates of all inflection points of or write DNE.
(ii) A cruise line offers a trip for $1000 per passenger. If at least 100 passengers sign up, the price is reduced for all passengers by $5 for every additional passenger (beyond 100) who goes on the trip. The boat can accommodate 250 passengers.
The number of passengers which maximizes the cruise line’s total revenue is
What price does each passenger pay if that number of passengers goes on the cruise?
(iii) Find the dimensions of the right circular cylinder of maximum volume that can be placed inside a sphere of radius R:
(iv) A certain tank consists of a right circular cylinder with hemispherical ends. For a given surface area S, find the dimensions (radius and length) of the tank with maximum volume (your answer should include S):
(v) A square piece of tin 24 in on each side is to be made into an open-top box by cutting a small square from each corner and bending up the flaps to form the sides. What is the side length of the square that should be cut from each corner to make the volume of the box as large as possible?
The average rate of change of the altitude of the roller coaster on the interval is .
Select the best interpretation of for .
Because is continuousdifferentiable on the interval and is continuousdifferentiable on the interval , satisfies the conditions of the Mean Value Theorem.
By the Mean Value Theorem, there exists in such that . In face this happens twice, when and when (assume ).
The steepest point on the roller coaster is . (Hint: maximize on .)
The linear approximation, , to at is
Using this linear approximation we estimate that is approximately This estimate is an overestimateunderestimate because is concave upconcave down between 5 and 7.
When changes from to the change in , is We can approximate this change with the differential which is
(i) At what point(s) does the conclusion of the Mean Value Theorem hold for on the interval ?
(ii) The equation of the line that represents the linear approximation to the function at is
(iii) Determine the following indefinite integral
(iv) Evaluate the expression
(a) What is the average rate of change in the population during the time interval ?
(b) Assume that is continuous on and differentiable on . Is there a moment in which the instantaneous rate of change of is equal to the average rate of change computed in part (a)?
If yes, select the theorem which guarantees the existence of such a point. If no, select ‘No Theorem’.
(ii) Let . The graph of is given in the figure below.
(a) The linear approximation L to the function at is
(b) Select the figure which includes the graph of :
(c) Use the linear apprxoimation to estimate the value of .
(d) The approximation in part (c) is an overestimate underestimate because is concave upconcave down .
(a) Calculate and the grid points (in ascending order)
(b) Select the figure with the right Riemann sum drawn in
(c) Calculate the right Riemann sum:
(i) Given the interval , what is ?
(ii) What is ?
(iii) Given that , select the expression for the grid points , for ?
(iv) What is for .
(v) Evaluate this Riemann sum
(vi) This Riemann sum is
(a)
(b)
(c)
(d)
(e)
(f)
Express the limit as a definite integral.
(i) Evaluate the following integrals
(a)
(b)
(c)
(ii) Assume that is odd. Evaluate
(ii) Assume that is even. Evaluate
(a) Find the velocity at time t:
(b) Find the distance traveled during the given time interval:
(c) Find the position at time t:
(a) Find the velocity at time t:
(b) Find the position at time t:
(a) Use geometry to evlaluate
(b) Select the graph of a rectangle whose net area is equal to
(ii) The graph of on the interval is given in the figure
(a) Use geometry to evaluate
(b) Compute