You should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises.

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1 BEFORE You Begin Calculus II If it has been awhile since you ha Calculus, I strongly suggest that you refresh both your ifferentiation an integration skills. I woul also like to remin you that in Calculus, as a mathematics course, connecting your work an correct notation are part of any correct solution. So when you review, always practice in the manner that you will eliver on exams, that is, write mathematics. Differentiation You are require to know the ifferentiation formulas from calculus I in the table below), as well as the prouct rule, quotient rule, chain rule, an the various properties of erivatives. You will not have a formula sheet available on quizzes or exams with these formulas. For erivative practice, go to my web-page, math.mercyhurst.eu/ griff/courses/m70/hw.php; you shoul work, as necessary, the exercises from sections:.3,.4,.5,.6, 3., 3.3 in the text Calculus - Early Transcenentals by Anton prouct an quotient rules, chain rule, erivatives of logarithmic, exponential, an inverse trigonometric functions). You shoul also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises. For further practice, I might suggest also working problems as necessary) from the attache Derivative Problems worksheet. If you wish to see more than the provie answers, follow the link: Derivative Problems, the file SOLN erivatives.pf which has scanne copies of the worke solutions. Integration You are require to know both the erivative an integration formulas from calculus I in the table below), the metho of substitution for inefinite an efinite integrals using correct notation), the Funamental Theorem of Calculus Pt ), along with the various properties of integrals. You shoul work, as necessary, the exercises from sections: 5., 5.3, 5.5 & 5.6 light), 5.9 complete); from the HW. Differentiation Formulas Integration Formulas. [x] = = x + C [ ] x r+. = x r x r = xr+ + C r ) r + r + 3. [sin x] = cos x cos x = sin x + C 4. [cos x] = sin x sin x = cos x + C 5. [tan x] = sec x sec x = tan x + C 6. [sec x] = sec x tan x sec x tan x = sec x + C 7. [ln u] = u u = ln u + C u u 8. [eu ] = e u u e u u = e u + C [ 9. sin u ] u = u a u u = u sin a + C 0. [ u ] u a tan = a a + u a + u u = a tan u a + C calc 6.vi

2 What is the concept Limit? It is reasonable to claim that the founation for all of calculus is inee the concept limit. Our notation for a limit: lim fx) = L The Relationship Between One-sie an Two-sie Limits: Limits we Nee: lim k = k lim x = c lim x = or lim x x xn = for n =,, 3, 4,... lim = 0 or lim = 0 for n =,, 3, 4,... x x x xn lim fx) = L lim fx) = L = lim fx) + Properties: Let c be any real number, k a constant, an suppose f, g have limits that exist at c. Then: You can rag constants through limits lim kfx) = k lim fx) The limit of a Sum/ifference [ ] lim fx) ± gx) = lim fx) ± lim gx) The limit of a Prouct ) ) lim fx)gx) = lim fx) lim gx) The limit of a Quotient fx) lim fx) lim gx) = lim gx), if lim gx) 0 The limit of a Power - generalize below [ ] [ ] r r lim fx) = lim fx) where r is a real number. If r is negative, then lim fx) 0. An important Theorem: If lim gx) = L an if the function f is continuous at L, then ) lim fgx)) = f lim gx) This states that limits can be ragge through continuous functions. The Squeeze Theorem Let f, g an h be functions satisfying gx) fx) hx) for all x in some open interval containing the number c, except possibly at c. If lim gx) = lim hx) = L then f has the same limit, lim fx) = L

3 What is the average change? The average rate of change of a function fx) between two values x = a an x = b is given by the quotient fb) fa))/b a). Geometrically, the average change is the slope of the secant) line passing through the points a, fa)) an b, fb)). If one uses the points x, fx)) an x+h, fx+h)), where h is an arbitrary constant, the expression for the average change becomes fx + h) fx) average change =. h Alternatively, one can write the average change y x The limit of ifference quotients. fx + x) fx) =. y slope =f x) The instantaneous rate of change of a function. The slope of a line tangent to the graph of a function. The best linear approximation to a function. Various rules an tables for computing. y y+ y y y=fx) The erivative f x) is the instantaneous rate of change of f at x. Geometrically, the erivative is the slope of the line tangent to the graph of y = fx) see the figure). For a function y = fx), we take as the efinition of the erivative x x x+ x Letting h = x, the efinition can be written f fx + h) fx) x) = lim. h 0 h = lim y x 0 x = lim x 0 fx + x) fx). x The expressions an are calle ifferentials an are relate to x an y as follows. In general, y. x the change in x, an x = y = the change in y = fx + x) fx) = f x) x The sign of the erivative etermines whether a function is increasing, ecreasing, or neither. If f x) > 0 then f the y value) is increasing, an if f x) < 0 then f is ecreasing. If f x) = 0 then f is neither increasing or ecreasing.

4 What is the integral? The limit of Riemann sums. The area uner the graph of a function. An antierivative. Various rules an tables for computing. A function F x) is an antierivative of the function fx) on an interval I, if F x) = fx) for all x in I. The inefinite integral fx) is the antierivative of f with respect to x, that is, fx) = F x) + C where C is an arbitrary constant calle the substituting F x) for fx) we obtain constant of integration. Integration is the inverse of ifferentiation, by F x) = F x) + C. Moreover, if fx) = F x) + C, then [ ] fx) = fx). Geometrically, the integral is the collection of graphs F x) + C whose tangent lines have slopes given by fx). 4 3 y x The figure shows the graphs of several antierivatives of / = 3x, that is, the graphs y = 3x ) = x 3 x + C for various values of C. The expression x 3 x + C is calle the general antierivative of 3x. The particular antierivative that satisfies the conition y) = 4 is y = x 3 x the bol curve in the figure). The integral b fx) is calle a a lines x = a an x = b. efinite integral an computes the net signe area between f, the x-axis, an the The Funamental Theorem of Calculus Part I): If f is continuous on [a, b] an F is any antierivative of f on [a, b], b a fx) = F b) F a) Integration by Substitution Suppose that F is an antierivative of f an that g is a ifferentiable function. To integrate fgx))g x) ) we make the substitution so that eq?? becomes: u = gx) an u = g x) fu) u = F u) + C The term inefinite integral is a synonym for antierivative.

5 Derivative Problems Fin the erivative of each of the given functions.. y = x + 4x + 6) 5. ft) = t t 5) 4 3. ht) = t ) 3/ 4. y = t 5 x 5. Gx) = 3x ) 0 5x x + ) 6. y = x 4) 4 8x 4) 3 7. y = x + ) 3 x + 8. y = sec x + tan x 9. Ry) = y 3y + ) 0. fx) = 3x ) 4x) /. gx) = x 4 x +. hx) = 5 3x x) 4 3. fx) = sin x sin 3x) 9 4. y = + sin x x + cos x 5. fx) = cos x sin x sin x + tan x) 6. gx) = x4 + 3x x 7. st) = 4 t3 + t 3 9. hx) = 8. fz) = x 7 3x 0. fx) = 5 z x 3/4 + 5x /6). y = cos a3 + x 3 ) 3 3. y = tan x sec x + t. s = t 4. fx) = ) 7 tan x sin 6 x + sin 4 x cos x 5. hx) = x + x + 9) /) / 7. y = x + x 8. R = t rt) = 3 7t t + t cos x ) tan x ) 3 ) x 9. fx) = sec x 30. y = ) 3x 4) /3 3. y = e k tan x 3. y = ln tan x) 33. y = + xe x 34. fx) = + ln x ) e x 35. y = cos + e x 36. rθ) = tan cos θ) ) 37. ft) = sin e sin t 38. y = tan x ) x +

6 Fin the secon erivative of each of the given functions. 39. gx) = x + x 4. hx) = 3x x fy) = 4. y = y y 3 5 x ) 3/4 43. If f an g are ifferentiable functions such that f) = 3, f ) =, g) = 5, an g ) =, fin the following values. a) f + g) ) b) 4f) ) c) fg) ) ) ff) ) ) ) e) f+g ) f) 5 g ) 44. Given the following table of values, fin the inicate erivatives. x fx) f x) gx) g x) a) h ), where hx) = fgx)) b) H ), where Hx) = gfx)) c) F ), where F x) = ffx)) ) G 3), where Gx) = ggx))

7 Solutions:. y = 0x + 4x + 6) 4 x + ). f 8 t) t) = t t 5) 5 3. h t) = 3 t ) / + t ) t 4. = 5 5 x 7 5. G x) = 303x ) 9 5x x + ) + 0x )3x ) 0 5x x + ) = 63x ) 9 5x x + ) 85x 5x + 9) 6. y = 8x 4) 3 8x 4) 3 48xx 4) 4 8x 4) 4 7. = x 3 ) x x + + x + )x + ) / = 4 sec x tan x 9. R y) = y + 3) 3y + ) 3 0. f x) = 5x 3/. g x) = x 4 x + ) 4x 3 x). h x) = 3. f x) = 9sin x sin 3x) 8 cos x sin 3x + 3 sin x cos 3x) 4. 83x ) 53x x) /5 = x cos x x + cos x) 5. f x) = sin x 6. g x) = 4x + x 3 7. s t) = t 3 ) 3/4 + 3t t 3 t 3 ) 8. f z) = z ) 6/ h x) = x 7 3x) 3/ 0. f x) = = x sin a 3 + x 3 ). = cos x + sin x h x) = 7. s t = 7 x + x + 9) /) / x + xx + 9) / ) 6. r t) = = x + + ) x x 9. f x) = x sin 5 x ) cos x ) = = = x 3/4 + 5x /6) 3 + t t f x) = sec x tan x 8. k x sec x) e k tan x 3. x e x + xe x 4e x ) e x + e x ) sin + e x 37. f t) = sin t) sin e sin t ) e sin t ) ) 37) from: 4 sin t cos t sin e sin t cos e sin t e sin t ) 6 4t t ) ) /3 t t 7t ) R t = t t ) = x 8 3x 4) 4/3 = sec x) ln tan x) tan x 34. f x) = x + ln x) 36. r θ) = sin θ + cos θ 38. = x + ) x 38) from: x + + x x + ) x /4 5 6 x 7/6 )

8 39. g x) = 6x ) f y) = 3y y ) 5/ 4. h x) = 6xx + 7) 5/ 4. y = 45 + x ) 5 x ) /4 43. a) b) 4 c) ) 6 e) 4 f) a) h ) = 30 b) H ) = 36 c) F ) = 0 ) G 3) = 63 calc 6.vi