Introduction to Calculus Practice Final
Introduction to Calculus Practice Final
Extrema on an Interval
1. Find all critical numbers:[pic]
2. Find all critical numbers: [pic]
3. Find all extrema in the interval [pic]if[pic].
4. Find the absolute maximum and absolute minimum of f on[pic].
[pic]
5. Explain why [pic] has a minimum on the interval [pic]but not on the interval[pic].
Rolle’s Theorem and the Mean Value Theorem
6. Consider the function[pic]. State why Rolle’s Theorem does not apply to f on the interval[pic].
7. Decide whether Rolle’s Theorem can be applied to [pic]on the interval[pic]. If Rolle’s Theorem can be applied, find all value(s) of c in the interval such that[pic]. If Rolles Theorem cannot be applied, state why.
8. Determine whether the Mean Value Theorem applies to [pic] on the interval[pic]. If the Mean Value Theorem can be applied, find all values of c in the interval such that[pic]. If the Mean Value Theorem does not apply, state why.
Increasing and Decreasing Functions and the First Derivative Test
9. Find all open intervals on which [pic]is decreasing.
10. Which of the following statements is true of[pic]?
(a) f is decreasing on (-3, -1)
(b) f is increasing on (-3,-1)
(c) f is increasing on [pic]
(d) f is increasing on [pic]
11. Find the values of x that give relative extrema for the function[pic].
12. Use the first derivative test to find the x-values that give relative extrema for[pic].
Concavity and the Second Derivative Test
13. Find all intervals on which the graph of the function is concave upward:
[pic].
14. Find all points of inflection: [pic].
15. Let [pic]and let [pic]have critical numbers -1, 0, and 1. Use the Second Derivative Test to determine which critical numbers, if any, give a relative relative maximum.
16. The figure given in the graph is the second derivative of a polynomial function, f. Sketch a possible graph for f. Make your x-axis from -3 to 3 and your y-axis from -3 to 3.
Optimization
17. The product of two numbers is 588. Minimize the sum of the first and three times the second.
18. A rancher has 300 feet of fencing to enclose a pasture bordered on one side by a river. The river side of the pasture needs no fence. Find the dimensions of the pasture that will produce a pasture with a maximum area.
19. An open box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimensions 6 inches by 6 inches.
20. A right circular cylinder is to be designed to hold 22 cubic inches of a soft drink. The cost for the material for the top and bottom of the can is twice the cost for the material of the sides. Let r represent the radius and h the height of the cylinder.
a. Write the equation for the surface area, SA, in r and h.
b. Write the cost function C.
c. Write the cost function as a function of one variable, r.
d. Find the radius that minimizes cost.
Other Applications of the First and Second Derivative
Related Rates…see Ch 2.6 Exam
Things you need to know:
Mathematical interpretation of velocity, speed, acceleration.
Finding limits (I will have a few limit problems on the exam…see CH 1 Exam)
Determining if a limit exists
Evaluating a Limit
One-sided limits (see guided resource CH 1)
Infinite Limits (see guided resource CH 1)
Do a few odd number problems from CH 1 regarding limits to prepare.
Note to student: I will have solutions for the problems on this study guide by Monday morning. Check back to check your answers and make corrections. On Tuesday I will answer any final questions, and have a few problems ready to help you rehearse for the final.
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