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Math 124 - Midterm 2 - Main Topics for Review

This midterm will be based mostly on the material from Chapter 3, especially sections: 3.4, 3.5, 3.6, 3.9.

It may also include parametric curve questions based on section 10.1 and the tangents part of section 10.2.

Rules of Differentiation:

It is important to know well and be able to apply correctly all the rules of differentiation (alone AND in combinations), including those from earlier in the quarter:

3.1: Constant Alone, Power Rule, Constant Multiple, Sum/Difference and Exponential Function Rule 3.2: Product Rule and Quotient Rule

3.3: Trig functions: sin(x) , cos(x) , tan(x) , cot(x) , sec(x) , csc(x)

3.4: Chain Rule

3.5: Inverse Trig Functions: arcsin(x) , arccos(x) , arctan(x) , arccot(x) , arcsec(x) , arccsc(x) 3.6: Derivatives (and properties) of Logarithmic Functions (ln(x) and loga(x))

Special methods of differentiation:

3.5: Implicit Differentiation (What is this method? When should you use it? Can you apply it reliably?) Some applications:

a)    Computing slopes of tangent lines to curves

b)    Finding formulas for derivatives of inverse trig functions (3.5)

c)    Finding formulas for derivatives of logarithms (3.6)

3.6: Logarithmic Differentiation (What is this method? When should you use it? Can you apply it reliably?)

Applications of differentiation:

3.9: Related Rates of Change: The general strategy is listed in your textbook and in lecture notes. Practice by doing many problems. Each problem is different, so it’s essential to read the problem carefully and do the set-up slowly    and very clearly! Be able to recognize and use: similar triangles, Pythagorean Theorem, ALL trig functions in a right  triangle, law of cosines. Volume/area formulas will be provided as needed (except for the most basic ones, such as  rectangles, circles, triangles, cubes). Remember to differentiate with respect to time!

Parameterized curves

10.1: Understand what a parameterized curve represents; how the two parametric equations relate to the curve itself; vertical and horizontal velocities; how to parameterize uniform linear motion; how to write down and

recognize the parameterization of a uniform circular motion along any circle; types of curves like in your homework; be able to eliminate the parameter and find the Cartesian equation of the curve.

10.2: Compute and use slopes and equations of tangent lines to parametric curves. Recall that d(d)x(y)  =    = x(y)′(′)t(t)

Also, horizontal & vertical velocities  d(d)t(x) = x′ (t), dt(dy) = y′ (t), and the speeds(t) = √(x′ (t))2  + (y′ (t))2

Review: Lectures, Homework, Worksheets

Pay attention to recurring themes, for example: equations of tangent lines (through points on or off the curve), parametric equations, circular motion (sin/cos), solving equations involving trig functions.

Practice:

•    There is an archive of sample exams online, under the Math 124 Materials website, Week 8, Midterm #2. Work through a couple of those in an exam-like setting (50-80 minutes, no help, no answers) and bring

questions to our review session. You may skip any problems referring to linear approximation, tangent line approximation, or to finding min/max of non-trigonometric functions.

•    There are also lots of practice problems in the review section at the end of Chapter 3 in your textbook, and two extra sample midterms I posted.