Calculus Spring Final Study Guide
Benzel
Final Exam INFO:
Your final will be cumulative with the start of the semester. These questions will give
you a good idea of what will be on the final and represent what we have gone through
in chronological order. Expect an exam that is designed to last 1-1.5 hours.
GETTING HELP:
Aside from my Tuesday morning office hours, I will be staying after school every day
next week. Feel free to let me know that you are coming in and I would be glad to help
you study for your final exam.
Suggested Vocabulary That You Should be Comfortable With:
- Implicit differentiation – Linear Approximation- Critical Point-Inflection Point
-First Derivative- Second Derivative- Maximum- Minimum- Concave Up/Down
- Rolle’s Theorem- Mean Value Theorem- Extreme Value Theorem- AntiderivativeIntegral- Fundamental Theorem of Calculus- Riemann Sum- Left Point ApproximationRight Point Approximation- Midpoint Approximation- Definite IntegralProperties of Integrals
I would suggest that you could make flashcards for these. These will help you with the
conceptual questions.
Friday:
Nothing Due Yet
However, please bring it to
class to begin work time on
it.
STUDY GUIDE DEADLINES:
Tuesday
Thursday
COMPLETE Pages 2-9 for a 3 Turn in the entire study
checkpoints
guide. We will grade on
completion.
BEST OF LUCK AND HAPPY STUDYING!
1
(1) Implicitly differentiate the following to find dy/dx
(a) 6𝑥 = 3𝑦 4 + 4𝑦
(b) 1 = 4𝑥 − 2𝑥 2 𝑦 2
(c) 𝑠𝑖𝑛𝑥 − 𝑐𝑜𝑠𝑦 = 1
(2) The radius of a circle is increasing at a rate of 2 inches per minute. Find the rate of change of area
when the radius is 6 inches.
(a) What are the important variables and relevant equations to solve this problem.
(b) Implicitly differentiate your area formula to find the rate of change of area when the radius is 6
inches
2
(3) Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which
the radius of the balloon is increasing when the diameter of the balloon is 20 cm. Implicitly
differentiate to determine the rate at which the radius of the balloon is increasing. (HINT: You may use
that the formula for the volume of a sphere is 𝟒/𝟑𝝅𝒓𝟑)
(4) (a) Find the local linear approximation of of 𝑓(𝑥) = 5𝑥 3 − 3𝑥 2 + 2 at the point where x = 0.
(b) Use your approximation to estimate f(-0.1), f(0), f(0.1)
3
(5) (a) Find the local linear approximation of 𝑓(𝑥) = √𝑥 at the point where x=64
(b) Use your approximation to estimate √65
(c) Use your approximation to estimate √63
(6) Fill in the following for the blanks.
(a) When f’(x) is positive, then the graph of f(x) is _____________________
(b) When f’(x) is negative, then the graph of f(x) is ____________________
(c) When f’’(x) is positive, then the graph of f(x) is ____________________
(d) When f’’(x) is negative, then the graph of f(x) is ____________________
(e) What does it mean when f’(x)=0?
(f) What does it mean when f’’(x)=0
4
(7) Consider the function𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 4
(a) Find the critical points of the function. Be sure to state which points are the local maxes and mins
(b) List the intervals for which the function is increasing/decreasing
(c) Find the inflection point(s) of the function
(d) List the interval for which the function is concave up/concave down
(7) Pictured below is a function, f(t). Complete the chart below indicating the sign (+ or – or 0) for
s(t), s ' (t ) and s' ' (t ) at each of the indicated points.
5
(8) . The graph of a twice differentiable function is shown below. Order the values of f(2), f ' (2)
and f ' ' (2) in order from least to greatest. Explain your reasoning by discussing such concepts as
instantaneous slope and concavity.
(9) In your own words, describe the difference between a local maximum and an absolute maximum.
6
(10) Matching. Put the appropriate Roman numeral number for the name of the Theorem
I. if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a
maximum and a minimum, each at least once.
II. If f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least
one number c in the interval (a,b) (that is a < c < b) such that
III. any real-valued differentiable function that attains equal values at two distinct points must have a
stationary point somewhere between them—that is, a point where the first derivative (the slope of the
tangent line to the graph of the function) is zero.
(a) Rolle’s Theorem ______________
(b) Extreme Value Theorem __________
(c) Mean Value Theorem___________
(11) Compare and contrast the extreme value theorem and the mean value theorem using the
matching that you did in the previous question.
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(12)
(a) Determine the value(s) for x in which the extreme value theorem applies for the function 𝑓(𝑥) =
4𝑥 3 + 15𝑥 2 − 18𝑥 + 7 on the interval [-1,4]
(b) Find the absolute minimum and absolute maximum in the interval [-1.4]
8
(13) Find the values of x for which Rolle’s theorem can be applied.
(a) 𝑓(𝑥) = 𝑥 2 − 5𝑥 + 4 on the interval [0,5]
(b) 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 𝑥 − 1 on the interval [-1,2]
(14) Find the value for x in which the mean value theorem is satisfied for the function
𝑓(𝑥) =
1
2
𝑥 2 − 2𝑥 − 1 on the interval of [-1,1]
9
(15)
Compute the antiderivatives of the following functions below:
(a) f ( x)  2 x  4
(b) f ( x)  3 x 
2
2
(c) f ( x)  3x  10 x  7
2
(d) . f ( x) 
4
3
 4
5
x
x
10
(16) Consider the following area problem
(a) Express this area problem as a definite integral to describe the area that you are computing?
(b) How could you approximate the area under the curve. You may use the graph to aid in your
explanation. Write out the procedure in which you would estimate the area as well as potential
limitations for your estimation.
(c) How could you come up with a new and improved estimation?
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(17)
Given below is a table of function values of h(x). Approximate each of the following definite integrals using the
indicated Riemann or Trapezoidal sum, using the indicated subintervals of equal length.
x
–3
–1
1
3
5
7
9
h(x)
2
6
4
8
-9
10
12
1
1.
 h( x)dx
9
using two subintervals and a Left
2.
3

Hand Riemann sum.
9
h( x ) dx using three subintervals and a
3
Midpoint Riemann sum.
using three subintervals and a Right
3
Hand Riemann sum.
3.
 h( x)dx
3
4.
 h( x)dx
using three subintervals and a
3
Left Point sum.
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(18) Evaluate the definite integrals in the space below:
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(19) Evaluate the following using the properties of integrals:
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