Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.)

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1 (1) (Pre-calculus Review Set Problems 80 an 14.) (a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample. (i) If a an b are real numbers an a 5 b 7 =3ab,thena 4 b 6 =3.. (ii) When solving x (x ) 3 =(x ) 3,wegetx =1,sothesolutionsare x 1 =1,x = 1.. (b) Simplify an write the following expression without negative exponents. Show your work. 6 1 r 3 r r 5

2 () Fin the erivatives of the given functions. You must show your work. But you o not have to simplify your answers. (a) z(t) = p t (t + t +5) (b) h(x) = 5x x p x +1 7 (c) (6 points) Fin y/x implicitly: e xy = x + y

3 (3) (a) (9 points) The graph below is the erivative of a function f(x), f 0 (x). Answer each of the following questions. You o not nee to explain. f (x) y x 4 (i) f(x) isincreasingontheinterval(s) (If you are not sure about the coorinates of the en points, an estimate will o.) (ii) f(x) isconcaveupontheinterval(s). (iii) The function f(x) achievesitsabsolutemaximumvalueontheinterval[ 4, 4] at x =. (b) (6 points) Write a parameterization for the line segment from (, 5) an ( 1, 3).

4 8 < x 1, 0 apple x apple (c) (6 points) Consier the function: f(x) =. : 3 x, <xapple 4 (i) Sketch the graph of f(x). Please label clearly the value of f at x =0, 1,, 3, an 4. (ii) Fin the average value of f(x) ontheinterval[0, 4]. () (6 points) Fin the equation of the tangent line (in the form of y = mx + b) atthe point (, 1) to the curve efine by the parametric equations x =t, y = t 3.

5 (4) Fin each of the following limits when it exists, write DNE otherwise. Show your work. (a) lim x! x 4x 4 +3x 4 x 5x 5x 4 x a (b) lim p x!a + x a,wherea>0. (c) lim x!1 (ln(x)) x

6 (5) Consier f(x) =xe x. Answer each of the following questions. You must show all work. (a) Fin f 0 (x) anallcriticalpoint(s)off(x). (b) Determine the interval(s) where f(x) is increasing an ecreasing. State the x- coorinate(s) of the point(s) where f achieves it local maximum or/an local minimum. (c) Fin f 00 (x) anallinflectionpoint(s)off(x).

7 (6) The figure below shows the curve y = p x,anarectanglewithitsupper-leftcorneron the curve, its sies parallel to the axes, its left en at x = a, anitsrightenatx = b. Let b be fixe as b = 0. Fin the value of a such that the rectangle has the maximum possible area. What is that maximum possible area? Show your work an give exact values.

8 The erivative of a function Useful formulas f 0 f(x + h) f(x) (x) =lim h!0 h Some rules of i erentiation x (cf(x)) = cf 0 (x) x (f(x)g(x)) = f 0 (x)g(x)+f(x)g 0 (x) f(x) = f 0 (x)g(x) f(x)g 0 (x) x g(x) g(x) x f(g(x)) = f 0 (g(x))g 0 (x) Di erentiation formulas x (xn )=nx n 1 x (ex )=e x x (ln x) = 1 x x (arcsin(x)) = 1 p 1 x (sin(x)) = cos x x x (tan(x)) = sec x (sec(x)) = sec x tan x x x (arccos(x)) = 1 p 1 x x (ax )=(lna)a x x (cos(x)) = x (cot(x)) = x (csc(x)) = sin x csc x csc x cot x x (arctan(x)) = 1 1+x (sinh(x)) = cosh(x) x (cosh(x)) = sinh(x) x x (tanh(x)) = 1 cosh (x) The linear approximation of a function f at a is given by f(x) f(a)+f 0 (a)(x a) Derivative of the inverse function If f is a one-to-one i erentiable function with inverse function f 1 an f 0 (f 1 (a)) 6= 0,thentheinversefunctionoff is i erentiable at a an x f 1 (a) = 1 f 0 (f 1 (a)).

9 Parametric Equations for a straight line: An object moving along a line through the point (x 0,y 0 ), with x/t = a an y/t = b has parametric equations x = x 0 + at, The slope of the line is m = b/a. y = y 0 + bt. The instantaneous spee of a moving object is efine to be s x y v = +. t t The quantity v x = x/t is the instantaneous velocity in the x-irection; v y = y/t is the instantaneous velocity in the y-irection. The velocity vector! v is written! v =!! v x i + vy j. For parametric curves, y Slope of curve = y x = y/t x/t ; y x = t x. x t Funamental Theorem of Calculus: Iff is continuous on the interval [a, b] an f(t) =F 0 (t), then b a f(t) t = F (b) F (a). The average value of a function f on an interval [a, b] isequalto 1 b a Comparison of Definite Integrals: Iff is continuous an m apple f(x) apple M for a apple x apple b, thenm(b a) apple Basic integration formulas: b a f(x) x apple M(b a). b a f(x) x. 1. x n x = xn C, (n 6= 1). x =ln x + C n +1 x 3. e x x = e x + C 4. a x x = ax ln(a) + C 5. sin(x) x = cos(x)+c 6. cos(x) x =sin(x)+c 7. sec (x) x =tan(x)+c 8. csc (x) x = cot(x)+c 9. sec(x)tan(x) x =sec(x)+c 10. csc(x) cot(x) x = csc(x) +C x +1 x =tan 1 1 (x)+c 1. p x =sin 1 (x)+c 1 x