diff --git a/source/calculus/exercises/outcomes/AD/AD5/template.xml b/source/calculus/exercises/outcomes/AD/AD5/template.xml index a111c3887..0790b24f9 100644 --- a/source/calculus/exercises/outcomes/AD/AD5/template.xml +++ b/source/calculus/exercises/outcomes/AD/AD5/template.xml @@ -2,58 +2,157 @@

- For \displaystyle f(x) = {{f}} , identify the regions for which f(x) is increasing and decreasing (if any). Additionally, identify and classify all local extrema. + For \displaystyle f(x) = {{f}} , identify the open intervals for which f(x) is increasing and decreasing (if any). Additionally, identify and classify all local extrema.

-

- We have that \displaystyle f'(x) = {{fp}}. - When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. - When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. - When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There is a critical point {{cp1}} which is neither a max nor min. -

+ + +

+ + We have that \displaystyle f'(x) = {{fp}}. +

+
+ +

+ + When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. +

+
+ +

+ + When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. +

+
+ +

+ When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There is a critical point {{cp1}} which is neither a max nor min. +

+
+
-

- We have that \displaystyle f'(x) = {{fp}}. - When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. - When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. - When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There is a critical point {{cp1}} which is a local minimum and a critical point {{cp2}} is a local maximum. -

+ + +

+ + We have that \displaystyle f'(x) = {{fp}}. +

+
+ +

+ + When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. +

+
+ +

+ + When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. +

+
+ +

+ + When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There is a critical point {{cp1}} which is a local minimum and a critical point {{cp2}} is a local maximum. +

+
+
-

- We have that \displaystyle f'(x) = {{fp}}. - When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. - When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. - When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There are no critical points. -

+ + +

+ + We have that \displaystyle f'(x) = {{fp}}. +

+
+ +

+ + When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. +

+
+ +

+ + When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. +

+
+ +

+ + When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There are no critical points. +

+
+
-

- We have that \displaystyle f'(x) = {{fp}}. - When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. - When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. - When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There is a critical point {{cp1}} which is a local minimum. -

+ + +

+ + We have that \displaystyle f'(x) = {{fp}}. +

+
+ +

+ + When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. +

+
+ +

+ + When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. +

+
+ +

+ + When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There is a critical point {{cp1}} which is a local minimum. +

+
+
-

- We have that \displaystyle f'(x) = {{fp}}. - When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. - When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. - When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There are critical points {{cp1}} which is a local maximum and {{cp2}} which is a local minimum. -

+ + +

+ + We have that \displaystyle f'(x) = {{fp}}. +

+
+ +

+ + When x < {{cv1}}, f'(x) {{sign11}} and y is {{change1}}. +

+
+ +

+ + When {{cv1}} < x < {{cv2}}, f'(x) {{sign12}} and y is {{change2}}. +

+
+ +

+ + When {{cv2}} < x, f'(x) {{sign13}} and y is {{change3}}. There are critical points {{cp1}} which is a local maximum and {{cp2}} which is a local minimum. +

+
+
diff --git a/source/calculus/exercises/outcomes/AD/AD6/template.xml b/source/calculus/exercises/outcomes/AD/AD6/template.xml index 4581f3b0a..6c5f697fe 100644 --- a/source/calculus/exercises/outcomes/AD/AD6/template.xml +++ b/source/calculus/exercises/outcomes/AD/AD6/template.xml @@ -2,52 +2,152 @@

- For \displaystyle f(x) = {{f}} , identify the regions where f(x) is concave up and concave down (if any) as well as all inflection points. + For \displaystyle f(x) = {{f}} , identify the open intervals where f(x) is concave up and concave down (if any) as well as all inflection points.

-

- We have that \displaystyle f''(x) = {{fpp}}. - When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. - When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. - When {{cc2}} < x < {{cc3}}, f''(x) {{sign23}} and y is {{concave3}}. - When {{cc3}} < x, f''(x) {{sign24}} and y is {{change4}}. There are inflection points {{ip1}} and {{ip2}}. -

+ + +

+ + We have that \displaystyle f''(x) = {{fpp}}. +

+
+ +

+ + When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. +

+
+ +

+ + When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. +

+
+ +

+ + When {{cc2}} < x < {{cc3}}, f''(x) {{sign23}} and y is {{concave3}}. +

+
+ +

+ + When {{cc3}} < x, f''(x) {{sign24}} and y is {{concave4}}. There are inflection points {{ip1}} and {{ip2}}. +

+
+
-

- We have that \displaystyle f''(x) = {{fpp}}. - When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. - When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. - When {{cc2}} < x < {{cc3}}, f''(x) {{sign23}} and y is {{concave3}}. - When {{cc3}} < x, f''(x) {{sign24}} and y is {{change4}}. There are inflection points {{ip1}}, {{ip2}} and {{ip3}}. -

+ + +

+ + We have that \displaystyle f''(x) = {{fpp}}. +

+
+ +

+ + When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. +

+
+ +

+ + When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. +

+
+ +

+ + When {{cc2}} < x < {{cc3}}, f''(x) {{sign23}} and y is {{concave3}}. +

+
+ +

+ + When {{cc3}} < x, f''(x) {{sign24}} and y is {{concave4}}. There are inflection points {{ip1}}, {{ip2}} and {{ip3}}. +

+
+
-

- We have that \displaystyle f''(x) = {{fpp}}. - When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. - When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. - When {{cc2}} < x < {{cc3}}, f''(x) {{sign23}} and y is {{concave3}}. - When {{cc3}} < x, f''(x) {{sign24}} and y is {{change4}}. There is an inflection point {{ip1}}. -

+ + +

+ + We have that \displaystyle f''(x) = {{fpp}}. +

+
+ +

+ + When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. +

+
+ +

+ + When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. +

+
+ +

+ + When {{cc2}} < x < {{cc3}}, f''(x) {{sign23}} and y is {{concave3}}. +

+
+ +

+ + When {{cc3}} < x, f''(x) {{sign24}} and y is {{concave4}}. There is an inflection point {{ip1}}. +

+
+
-

- We have that \displaystyle f''(x) = {{fpp}}. - When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. - When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. - When {{cc2}} < x, f''(x) {{sign23}} and y is {{concave3}}. - There is an inflection point {{ip1}}. -

+ + +

+ + We have that \displaystyle f''(x) = {{fpp}}. +

+
+ +

+ + When x < {{cc1}}, f''(x) {{sign21}} and y is {{concave1}}. +

+
+ +

+ + When {{cc1}} < x < {{cc2}}, f''(x) {{sign22}} and y is {{concave2}}. +

+
+ +

+ + When {{cc2}} < x, f''(x) {{sign23}} and y is {{concave3}}. +

+
+ +

+ + There is an inflection point {{ip1}}. +

+
+