Final Exam Review
Supplemental Instruction
IowaStateUniversity / Leader: / Wheaton
Course: / Math 165
Instructor: / Dr. Tokorcheck
Date: / 02/24/15

Note: Headings are lecture titles. You may not actually use a stated theorem in the problems.

1)Lecture 18 Differentials. Solve the following problems using differentials.

  1. A boat is moving into shore at a rate of . The boat is observed from a lighthouse. What is the rate at which the distance between the boat and the observer changing when the boat is from shore?
  2. A hot air balloon takes off from ground level and rises at a rate of . There is an observer at ground level 100 ft away. How fast is the distance between the balloon and the man changing when the balloon has been rising for 20 seconds?
  3. A gobstopper (spherical) is being eaten such that the surface area changes at a rate of . At what rate is the volume changing when the radius of the gobstopper is 1 cm?
  4. One face of a cube is found to have a parameter of . What is the volume of the cube?

2)Lecture 19 Differentials and Approximations. Solve the following problems using differentials.

  1. The height of a cylinder is measured to be , while the radius is known to be . Find the volume of the cylinder and give an estimate for the possible error of this value.
  2. A cubic room of side length is to have applied a 0.04 inch coat of interior paint (include ceiling, do not include the floor). Estimate how many cubic feet of paint will be needed.
  3. Approximate
  4. Approximate
  5. Approximate

3)Lecture 20 Maxima and Minima. Find all local maxima and minima for the following functions. What is the global maxima and minima? Hint: first derivative test.

  1. on
  2. on
  3. on

4)Lecture 21 Concavity.Find the inflection points of the functions in the previous problem. Use these to determine when the function is concave up, or concave down. Hint: second derivative test.

5)Lecture 22 Graphing with Derivatives. Using the first and second derivative tests, sketch a graph approximating the graph of the functions in the previous two problems.

6)Lecture 23 Mean Value Theorem.Use the mean value theorem to determine if the following function can exist.

  1. Use the mean value theorem to show that has exactly one real root on

7)Lecture 24 L’Hôpital’s Rule. Use L’Hôpital’s Rule to evaluate the following limits.

8)Lecture 25 Optimization Problems. Solve the following optimization problems.

  1. Consider the production of cell phones, where the cost of production, revenue and profit are given below. What is the production that gives the highest profit? (All numbers are in USD).
  1. Consider the production of bubblegum, where the cost of production, revenue and profit are given below. What is the production that gives the highest profit? (All numbers are in USD).
  1. Consider a right cone inscribed inside a sphere or radius . Assuming the axis of symmetry of the cone passes through the center of the sphere, what is the maximum volume of the cone?

9)Lecture 26 Antiderivatives. Solve the following antiderivatives.

  1. A vehicle leaves a stoplight at a constant acceleration of . What is the distance traveled by this vehicle after 5 seconds?
  2. Evaluate the following sums:

10)Lecture 27 Antiderivatives II. Solve the following antiderivatives or determine the value of the summation.

  1. A vehicles is approaching a red light at . Safe breaking distance at this speed is given as 100 ft. Therefore, if breaking begins 100 feet from the light, what is the required acceleration to stop at the end of the 100 ft? Assume constant acceleration, and that it takes 4 seconds for the vehicle to reach a velocity of 0.

11)Lecture 28 Antiderivatives III. Solve the following antiderivatives using the Riemann sum, or from definitions. You may need arc functions.

12)Lecture 29 Integrals I.Solve the following:

  1. Given Riemann sum notation, give it’s equivalent definite integral
  2. Solve these integrals using geometric arguments:
  3. where
  4. Find the area under the curve:
  5. ,

13)Lecture 30 Integrals II. Solve the following definite integrals. Using the mean value theorem, determine the point on the curve which defines a rectangle of area equal to that under the curve. What is the average function value?

14)Lecture 31 Fundamental Theorem of Calculus.Solve the following definite integrals using the second fundamental theorem of calculus, or rate of accumulation problems with the first fundamental theorem.

15)Lecture 32 & 33 SubstitutionSubstitution II. Use u substitution to evaluate the following definite integrals.

16)Lecture 34 Integrating Trig Functions. Evaluate the following integrals using symmetry arguments if necessary. Hint: check if the function is odd or even.

17)Lecture 35 Area Between Curves. Find the area between the given curves.

18)Lecture 36 Logarithms and Exponential Functions. Find the following integrals or derivatives (you may need arc functions).

19)Lecture 37 Exponential Growth and Decay. Solve the following growth/decay problems.

  1. E. coli population doubles every 20 minutes under ideal conditions. When plating cultures, a biologists aims to spread a dilute solution of E. coli, such that each colony is started by a single cell. If the colonies are allowed to grow for 24 hours, what is the population? When is the population E. coli?
  2. A chemical reaction () takes place inside a vessel. The rate at which the concentration of A ( units of ) changes is given below as first order reaction:

If the initial concentration of A is , give the equation describing the concentration of A as a function of time.

20)Lecture 38 Exponential Growth and Decay II. Solve the following differential equations.

21)Lecture 39 Rates of Growth. For the following functions, compare their rate of growth. That is, which functions is of higher order?

22)Lecture 40 The Hyperbolic Functions.Dr. Tokorcheck has said that this will not be covered on the exam. Also, some sections did not cover this material.

23)Lecture 41 Exam Review.No new material was covered.