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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
Constant Rule: Power Rule:
Multiple Rule: Chain Rule:
Product Rule: Quotient Rule:
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.
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
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.
7. 8. 9. 10.
Concept: The Meaning of a Derivative • The derivative of a function represents the slope of all tangent lines to that function. • The derivative evaluated at a specific input represents the slope of the particular tangent line at the
point on the graph corresponding to that x-value. • More importantly, a derivative tells you how a function is changing! • If the sign of is positive, then is increasing, • If the sign of is negative, then is decreasing.
Remember: • If increasing the x-value causes the y-value to increase, this is positivechange. • If increasing the x-value causes the y-value to decrease, this is negativechange.
Example The graph of is a parabola. As you move from left to right, the slope of the parabola starts out very negative, becomes less negative, then zero, then positive, finally becoming more and more positive.
represents all of these slopes. Examples
When x is negative, the slope of the graph is negative.
As x increases, the slope also increases (it is less negative). When x = 0, the slope of the graph is 0. When x is positive, the slope of the graph is positive. As x increases, the slope also increases.
Concept: Second Derivative Since is the derivative of , it tells you how is changing.
• If the sign of is positive, then is increasing. • If the sign of is negative, then is decreasing.
Concavity • If a graph is above all of its tangent lines, we say the graph is concave up. This occurs when .
As it says above, being positive means is increasing. Therefore, when a graph is concave up, the first derivative of f is increasing.
• If a graph is below all of its tangent lines, we say the graph is concave down. This is when . Again, being negative means is decreasing. Therefore, when a graph is concave down, the first derivative of f is decreasing.
Back to the example… As you move along the parabola from left to right (read description above again), the slopeis always increasing. This means that is always increasing. Therefore, the graph of f is always concave up.
Some practice… 11. Given . Find the slope of the tangent to the graph of f at the point . 12. Find the equation of the tangent line to when x = 1.
13. Find the equation of the normal line to when x = 1.
14. At what point(s) on the graph of is the slope of the tangent line equal to 4?
15. 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.
• For the other angles, remember some basic right triangles:
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.
1. First, rewrite the function by ‘unadding’: Then, differentiate using the power and sum/difference rules: Finally, rewrite:
2. Use the product rule! Factor to simplify:
3. Use the quotient rule! Factor: Cancel common factor: OR
4. Use power rule: Simplify:
5. First ‘unadd’: [No quotient rule needed!!]
6. If you want, you can rewrite the power: Then use power rule: Simplify:
7. Direct substitution of 3 for x results in . So change the form (by factoring) and find the limit of this new function.
8. Direct substitution of 3 for x results in . So change the form (by rationalizing the numerator)…
9. 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,
10. Direct substitution results in . Therefore, .
11. First, . Then, . The slope of the tangent line to the graph of f at the given point is –2.
12. We need a point: Since , the point is (1, –4). We also need a slope: , so the slope = Finally, use point slope form:
13. The normal line is perpendicular to the tangent line, so it’s slope is . Again, using point slope form:
14. We want to know when the slope equals 4, so set the derivative equal to 4.
Still need the POINTS: and
15. We want to know when the slope of the tangent line is 0, so set the derivative equal to 0.
Still need the POINTS: (2, 5) and (4, 1)