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Basic review of calculus 1, Exercises of Calculus

Gannon UniversityCalculus

Review of Calculus 1 (Includes formulas and definitions)

Typology: Exercises

2015/2016

Uploaded on 05/18/2016

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BASIC REVIEW OF CALCULUS 1
Derivative Notation
You should be comfortable with and understand the differences between notation such as the following:
, , , and
Differentiation Rules
Constant Rule: Power Rule:
Multiple Rule: Chain Rule:
Product Rule: Quotient Rule:
Trig Rules:
NOTE: Calculus does NOT change the angles of trig functions!
Here area few problems for practice. Simplify all answers. An answer key is at the end of the review.
Differentiate.
1.
2.
3.
4.
5.
6.
pf3
pf4
pf5

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BASIC REVIEW OF CALCULUS 1

Derivative Notation You should be comfortable with and understand the differences between notation such as the following: , , , and

Differentiation Rules Constant Rule: Power Rule:

Multiple Rule: Chain Rule:

Product Rule: Quotient Rule:

Trig Rules:

NOTE : Calculus does NOT change the angles of trig functions!

Here area few problems for practice. Simplify all answers. An answer key is at the end of the review.

Differentiate.

Concept: The Limit of a Function

  • As the inputs of a function approach a number a, the outputs may or may not be approaching a number.

If the outputs DO approach a value…we write

which means you can make the output as close to L as you want just be inputting values close enough to a.

If the outputs DO NOT approach a value…the limit does not exist. When the limit does not exist, it may be due to the fact that the outputs keep increasing without bound instead of approaching some limit L. If this is true, we write. (Outputs could also be decreasing without bound, or neither could be occurring.)

  • As you input larger and larger numbers into a function, the output may or may not approach a number.

If the outputs DO approach a value…we write

which means you can make the output as close to L as you want just be inputting values LARGE enough.

If the outputs DO NOT approach a value…the limit does not exist. As above, the outputs could be increasing without bound , decreasing without bound , or neither.

  • To evaluate a limit, you can instead evaluate the function at that input IF the function is continuous

there.

means to just ‘plug in a’ to the function in order to evaluate the limit. (direct substitution)

  • If the function is not continuous, there are a few things to try:
  • If direct substitution results in , then the limit does not exist. You can then intuitively determine if the outputs are increasing or decreasing without bound ( or –), or if they aren’t (dne).
  • If direct substitution results in , you need to change the form of the function by factoring, rationalizing, or some other method, and then try to evaluate the limit of the new function.

IMPORTANT NOTE: is indeterminate because any number times 0 equals 0. is undefined because no number times 0 equals a (where ).

Here area few limits for practice. Simplify all answers. An answer key is at the end of the review.

Some practice…

  1. Given. Find the slope of the tangent to the graph of f at the point.
  2. Find the equation of the tangent line to when x = 1.
  3. Find the equation of the normal line to when x = 1.
  4. At what point(s) on the graph of is the slope of the tangent line equal to 4?
  5. Find all points on the graph of where there is a horizontal tangent line.

Trigonometry review It is expected you know the definitions of the trig functions:

It is also expected that you know the basic identities:

And it wouldn’t hurt to know a few more identities:

You should also know the trig values of a few angles (0, 30, 60, 45, 90).

  • For the quadrantal angles (0, 90, etc.), think about the graphs of sine and cosine.

, etc…

  • For the other angles, remember some basic right triangles:

etc… etc…

Don’t forget radians!! And reference angles!! 30 is the reference angle for 150, 210, 330, etc. The trig functions of these angles only differ (possibly) in sign.

For example…

ANSWER KEY

  1. First, rewrite the function by ‘unadding’:

Then, differentiate using the power and sum/difference rules: Finally, rewrite:

  1. Use the product rule! Factor to simplify:

Final answer:

  1. Use the quotient rule! Factor: Cancel common factor: OR
  1. Use power rule: Simplify:
  2. First ‘unadd’: [No quotient rule needed!!]

Now, differentiate:

  1. If you want, you can rewrite the power:

Then use power rule: Simplify:

  1. Direct substitution of 3 for x results in. So change the form (by factoring) and find the limit of this new function.
  2. Direct substitution of 3 for x results in. So change the form (by rationalizing the numerator)…
  3. Direct substitution of 3 for x results in. Therefore, the limit does not exist! We can further describe how the limit doesn’t exist… The numerator is approaching –1. The denominator is approaching 0 which causes the entire fraction to increase without bound. However, since x is approaching 3 from the right side (bigger than 3), squaring it produces a number always bigger than 9, and then subtracting 9 always leaves a small positive number.

Finally, since the numerator is always negative and the denominator is always positive AND going to 0,

  1. Direct substitution results in. Therefore,.
  2. First,. Then,. The slope of the tangent line to the graph of f at the given point is –2.
  3. We need a point: Since , the point is (1, –4). We also need a slope: , so the slope = Finally, use point slope form:
  4. The normal line is perpendicular to the tangent line, so it’s slope is. Again, using point slope form:
  5. We want to know when the slope equals 4, so set the derivative equal to 4.

Still need the POINTS: and

  1. We want to know when the slope of the tangent line is 0, so set the derivative equal to 0.