1. 1. Solution key to exercises 7.1 and 8.1 Chain Rule and Derivatives of Trigonometric Functions
2. 2. Exercises 7.1 Chain Rule Write your solutions orderly and neatly. Techniques to remember: Product rule and Chain Rule 4 15and4Let xxgxxh 5154',2' 32/1 xxgxxh 51544152''' 342/1 xxxxhgghf
3. 3. Exercises 7.1 Chain Rule 8 3324 2 12 212433312'' ' x xxxx g hggh f
4. 4. Exercises 7.1 Chain Rule Observe Chain Rule xxxxuuuuy 219216'9'6' 225225
5. 5. Exercise 7.1 Chain Rule The tangent line to the curve is horizontal means the derivative of the function at the point of tangency is ZERO 0TLm 0TLm Analysis by graphs has limitations
6. 6. Exercises 7.1 Chain Rule 042343' 22 xxxy 2042 xx 013 03434 22 xx xxxx 1,3 xx
7. 7. Exercises 7.1 Chain Rule Recall the equation of a line, in particular, the point-slope form 11 xxmyy 1,1, 11 yx 1at' xym
8. 8. Exercises 7.1 Chain Rule 2' ,1at 2 22 ' 2 2212 ' 2 2/122 2 2/12 2 12 y x x xx y x xxxx y Equation of line is 121 xy
9. 9. Exercises 8.1 Derivative Trigonometric Functions xxxxxf 105sec5tan1' 222 Chain rule xxxxf 211sec' 2/12 2 122
10. 10. Exercises 8.1 Derivative Trigonometric Functions 11 xxmyy 0,, 11 yx xym at' 24 2 4 22 1012tansec ' x xxxx y xy 2 1
11. 11. Exercise 8.1 Derivative Trigonometric Functions
12. 12. Exercises 8.1 Derivative Trigonometric Functions radians31.152.75cos 01cos4' 4 11 o x xy Solution set: The cosine function is a periodic function! It repeats the cycle every 2pi radians. integeranyis ,231.1 k kx
13. 13. Exercises 8.1 Derivative Trigonometric Functions 0tansec4tansecsec4)('' 1sec2' 2 2 xxxxxxf xxf Kelan nagze-zero ang second derivative? 0tanor0sec0tansec2 xxxx ,0x 180,0x