sin 2 x Lim x sin sin cos x x x Lim x - 2 1 1,0 ? f x x at = + -

Study Guide MAT220 Midterm
Syllabus approved calculators are okay to use on this assessment BUT correct handwritten work must accompany your answers if you
expect to earn points for a problem.
1. Find a limit using conjugation. Like H.W. 1.2 # 9, 10. Also see your class notes in this section for more examples.
2. Find a limit where you will have to utilize one of the special trig limits (you may also utilize some properties of limits in your
solution). See YOUR class notes.
x 0
sin 2 x
x 0
sin x  sin x cos x
3. A. Given a piecewise defined function, figure out whether or not it is continuous at a particular x – value.
B. Given a piecewise defined function, figure out whether or not it is differentiable at a particular x – value.
The topic of continuity can be reviewed by looking back through your 1.3 HW. To see a couple of problems that may be similar to
this see HW 1.5 #8 and 9. The difference in YOUR test question solution is that you may discuss the differentiability of the function
by utilizing the differentiation rules rather than the limit definition of derivative that we were restricted to back in section 1.5 (THIS
will make answering part B quicker).
4. A problem where you are asked to find the equation of the tangent line to a given function at a particular point USING either the
limit definition of derivative OR the alternate limit definition of derivative (is one of those easier here?). You can use H.W. 1.5 # 1
and 2 to practice for this problem.
Also, could you do it for the function
f  x   x  2  1 at
 1,0 ?
5. – 8. Some problems where you will be asked to use the various differentiation rules that we have covered (as opposed to the limit
definition of derivative) to find the derivative of some functions. You should review problems from the 1.6 H.W. on basic
differentiation rules along with the 2.1 H.W. on the product and the quotient rule and ALSO the 2.2 H.W. on the chain rule. You
should be sure that you also look through the 2.4 H.W. as one or more of these problems may contain an inverse trig function as part
of it.
Your HW and notes will provide many examples for you to go practice. You can also utilize the supplemental readings and videos on
the Calculus HELP page for more problems.
Here are some other problems to practice if you can’t find enough through the other resources.
A. y  e 4 x cos 2 x
B. y  ln  esin x 
C. y  ln  sin 2  4 x  
cot x
(do this one WITHOUT rewriting the function first!)
csc x
xe x
F. y 
G. y  arctan  sec x 
ln x
E. y 
H . y  arccos  sin x 
0 x
9. One problem involving implicit differentiation (H.W. 2.3)
10. One problem involving logarithmic differentiation (H.W. 2.3)
D. y   tan x  ln  x 2 