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g(x) = = ax + C, where a ER Power: is continuous on [a, b], differentiable on (a, b) and g'(x) = f(x). +C, where r + – 1 r +1 FUNDAMENTAL THEOREM OF CALCULUS II If fis continuous on [a,b] and F is any antiderivative of f, then Edx = In|x|+C Exponential: Sndx = F(b) – F(a). b +C, where b e (0,00) In(b) Trigonometric: NET CHANGE THEOREM cos(x)d x = sin(x) +C If F' is continuous on [a, b], then sin(x) = - cos(x) +C F'(x)dx = F(b) – F(a). Jsec"wnd x = tan(x) + C VARIABLE SUBSTITUTION sec(x)tan(x)d x = sec(x) +C If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Jescundx = - cot(2) + C csctr)cot(x)dx = - esc(x) + C VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b], then tan(x)dx = - In|cos(x)| +C Janterds cot(x)dx = In|sin(x)| +C f(u)du. v)dx = In|sec{x) + tan(x)|+ C ela) csc(x)dx = In|csc(x) - cot(x)| +C VOLUMES OF SOLIDS OF REVOLUTION Inverse Trigonometric: Revolving about x Revolving about y- axis dx aresin(x) + C, where x #± 1 -ахis Disks/Washers Integrate x variable Integrate y variable dx = arctan(x) + C Cylindrical Shells Integrate y variable Integrate x variable Нуpertbolic AVERAGE VALUE OF A FUNCTION cosh(x)d x = sinh(x) +C Iffis continuous on (a, b), then sinh(x) + C = cosh(x) +c favg f(x)d x.

g(x) = = ax + C, where a ER Power: is continuous on [a, b], differentiable on (a, b) and g'(x) = f(x). +C, where r + – 1 r +1 FUNDAMENTAL THEOREM OF CALCULUS II If fis continuous on [a,b] and F is any antiderivative of f, then Edx = In|x|+C Exponential: Sndx = F(b) – F(a). b +C, where b e (0,00) In(b) Trigonometric: NET CHANGE THEOREM cos(x)d x = sin(x) +C If F' is continuous on [a, b], then sin(x) = - cos(x) +C F'(x)dx = F(b) – F(a). Jsec"wnd x = tan(x) + C VARIABLE SUBSTITUTION sec(x)tan(x)d x = sec(x) +C If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Jescundx = - cot(2) + C csctr)cot(x)dx = - esc(x) + C VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b], then tan(x)dx = - In|cos(x)| +C Janterds cot(x)dx = In|sin(x)| +C f(u)du. v)dx = In|sec{x) + tan(x)|+ C ela) csc(x)dx = In|csc(x) - cot(x)| +C VOLUMES OF SOLIDS OF REVOLUTION Inverse Trigonometric: Revolving about x Revolving about y- axis dx aresin(x) + C, where x #± 1 -ахis Disks/Washers Integrate x variable Integrate y variable dx = arctan(x) + C Cylindrical Shells Integrate y variable Integrate x variable Нуpertbolic AVERAGE VALUE OF A FUNCTION cosh(x)d x = sinh(x) +C Iffis continuous on (a, b), then sinh(x) + C = cosh(x) +c favg f(x)d x.

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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FUNDAMENTAL THEOREM OF CALCULUS I If fis continuous on [a,b] then the function BASIC ANTIDERIVATIVES Constant = ax +C, where a ER %3D Power: is continuous on [a, b], dfferentiable on (a, b) and gʻ'(x) = f(x). (rds = +C, where r + - 1 r +1 FUNDAMENTAL THEOREM OF CALCULUS II If fis continuous on [a,b] and F is any antiderivative of f, then = In |x|+C Exponential: f(x)dx = F(b) – F(a). b +C, where b E (0,00) In(b) Trigonometric: NET CHANGE THEOREM cos(x)dx = sin(x) +C If F' is continuous on [a, b), then Jamer- sec-(endx = tan(x) + C = - cos(x) +C F(x)dx = F(b) – F(a). VARIABLE SUBSTITUTION sec(x)tan(x)d x = sec(x) + C If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Joscondx = - cot(2) + C r)cot(x)dx = - csc(x) +C VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS tan(x)dx = - In|cos(x)|+C_ If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Jcot(o)d x = In |sin(<)| + c f(u)du. sec(x)dx = In|sec(x) + tan(x)| + C gla) x)dx = In[cse(x) – cot(x)| +C VOLUMES OF SOLIDS OF REVOLUTION Inverse Trigonometric: Revolving about x Revolving about y- xp = arcsin(x) + C, where x *±1 -ахis axis Disks/Washers Integrate x variable Integrate y variable dx = arctan(x) + C 1+x2 Cylindrical Shells Integrate y variable Integrate x variable Нурerbolic AVERAGE VALUE OF A FUNCTION cosh(x)d x = sinh(x) +C Iffis continuous on [a, b], then sinh(x) +C = cosh(x) + C Savg = f(x)dx. b- a
Transcribed Image Text:FUNDAMENTAL THEOREM OF CALCULUS I If fis continuous on [a,b] then the function BASIC ANTIDERIVATIVES Constant = ax +C, where a ER %3D Power: is continuous on [a, b], dfferentiable on (a, b) and gʻ'(x) = f(x). (rds = +C, where r + - 1 r +1 FUNDAMENTAL THEOREM OF CALCULUS II If fis continuous on [a,b] and F is any antiderivative of f, then = In |x|+C Exponential: f(x)dx = F(b) – F(a). b +C, where b E (0,00) In(b) Trigonometric: NET CHANGE THEOREM cos(x)dx = sin(x) +C If F' is continuous on [a, b), then Jamer- sec-(endx = tan(x) + C = - cos(x) +C F(x)dx = F(b) – F(a). VARIABLE SUBSTITUTION sec(x)tan(x)d x = sec(x) + C If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Joscondx = - cot(2) + C r)cot(x)dx = - csc(x) +C VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS tan(x)dx = - In|cos(x)|+C_ If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Jcot(o)d x = In |sin(<)| + c f(u)du. sec(x)dx = In|sec(x) + tan(x)| + C gla) x)dx = In[cse(x) – cot(x)| +C VOLUMES OF SOLIDS OF REVOLUTION Inverse Trigonometric: Revolving about x Revolving about y- xp = arcsin(x) + C, where x *±1 -ахis axis Disks/Washers Integrate x variable Integrate y variable dx = arctan(x) + C 1+x2 Cylindrical Shells Integrate y variable Integrate x variable Нурerbolic AVERAGE VALUE OF A FUNCTION cosh(x)d x = sinh(x) +C Iffis continuous on [a, b], then sinh(x) +C = cosh(x) + C Savg = f(x)dx. b- a
10. Pictured below is the graph of a function F which is an antiderivative of the function f. Using the fact that it is an antiderivative, determine the following. a. List critical points of F b. f(1) с. F(x)dx d. f(x)dx e. Fis not continuous at x = - 1 and x = 3. What does this imply about f? 10
Transcribed Image Text:10. Pictured below is the graph of a function F which is an antiderivative of the function f. Using the fact that it is an antiderivative, determine the following. a. List critical points of F b. f(1) с. F(x)dx d. f(x)dx e. Fis not continuous at x = - 1 and x = 3. What does this imply about f? 10
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