. If f = e ta -, the f = e e p e e p e p+ 4 f = e ta -, so f = e ta - + e, so + f = e p + e = e p + e or f = e p + 4. The slope of the lie taget to the curve - + = at the poit, - is - 5 Differetiate - + = implicitl to get - - + = Now, plug i the poit, - to get + 4 - - 4 = = or = 5 D. If the fuctio f is cotiuous for all real umbers ad f = B, - a statemets must be true? I. f = B II. f = B III. f a = B a + a I ol II ol I ad II ol III ol I, II, ad III the which of the followig B the defiitio of cotiuit, f = B = f a ad all statemets must be true, a = f - - - 4 - - For problems 4 -, refer to the graph of = f, the derivative of f, show above. The graph cosists of five lie segmets, two of which are horizotal. 4. At =, f has a poit of discotiuit poit of iflectio poit of odifferetiabili local maimum local miimum Note that there is a corer at =, o the graph of f. Here, the secod derivative is chagig from - to, ad so there is a poit of iflectio at =, B 5. Over the iterval - 4 < < 4, how ma local miima does f have?
Oe Two Three Four Five Note that the derivative chages sig from - to + at = - ad =. Therefore, f has two local miima, B. If f =, what is the value of f -? - - 5 Cout them up! Area from to is -, ad area from - to is so f - = - - - remember that we ' re movig from left to right, so we subtract, ad so f - = C 7. Provide the most accurate approimatio for the followig defiite itegral : 4 - - d 7 4 9 Split the itegral ito two : 4 - - d = 4d - - d = - 4 p 4 = - 4 p ª - 4.4 =.44 B 8. = + = 8 9 for = + = + so a =, b =, ad d = or d = d = = 7 - = 9. The displacemet s t i meters of a particle movig o the - ais is a fuctio of time t i secods. The table below shows this displacemet for time values o the iterval, 8 t secods 5 7 8 s t meters 7 9 8 4 9 Usig the table above, provide the best approimatio for the particle' s speed at t = secods. - 4 meters secod - meters secod - meters secod Use a smmetric differece quotiet to approimate the velocit, v ª meters secod - 8 7-5 However, we are ased for the speed magitude of the velocit, so the aswer is D = - meters secod 4 meters secod = - meters secod. Give that f - = 4 ad f - =, which of the followig is the taget lie approimatio of f -.?
.8.9 4. 4. 4. The equatio of the taget lie is - 4 = - - or = + + 4 or = + ad = -. + = -. =.8 A. The slope field below matches which differetial equatio? 4 4 4 4 d = d = d = d = l d = si B ispectio, the aswer is A. csc = - csc = si = si = = B. If the differetial equatio the f = d = - has the solutio curve = f cotaiig the poit, 4, 4 d = - = - = The it does ot eist. - ad this matches the model for logistic growth, with a carrig capacit of M =, which represet the it as o the fuctio, C 4. If f + f = f, for all positive real umbers ad, which of the followig could defie f? e si 4 l l ab = l a + l b so l + l = l
5. The equatio of the lie taget to = + + at its poit of iflectio is = - - = - + = + = - = 4 + = + ad = + ad there is a PI at = -, where m = - ad the poit is -, 4 so - 4 = - + or = - + B. Give the graphs of h, f, ad g the followig could be true h f g h = f ad h = g h = g ad h = f f = g ad f = h f = h ad f = g g = f ad g = h Through guess ad chec, we see that f = h ad h = g Aother wa to epress this is f = h ad f = g The aswer is D 7. The side of a cube is epadig at a costat rate of cetimeters per secod. What is the istataeous rate of chage of the surface area of the cube, i cm per secod, whe its volume is 7 cubic cetimeters? 4 54 7 S = s so ds = cm cm dt sec ds = s ds dt dt = 7 cm sec Now, V = s so 7 = ad s = cm, so 8. The efficiec of a motor scooter egie is give b the cotiuous fuctio c, where is measured i gallos mile ad c is measured i miles. What are the uits of c dc? miles gallos gallo - miles gallos mile miles gallo The itegral represets the accumulatio of c i gallos mile over a iterval of miles. gallos i miles or gallos The aswer is B mile Therefore, the uits are 9. If f = e, which of the followig is equal to f e? e + h e + h - e e e e + h - e e e + h - e + h - e Usig the defiitio of the derivative, C e + h - e f = so e e + h - e e f e =
. Give that F = f, the value of f d b is F b - F a b F b - a F a a bf b - af a F b - F a F b - F a Let u =, ad du = d or du = d a b f u du = F u b = a F b - F a A. Cosider the graph of f below. If f 5 = -, the f = - f 4 5 - -.5 - -.5.5 The area from = to = is, ad the area from = to = 5 is -.5, so f = - +.5 - or f = -.5 A. Which of the fuctios give below has a average value of o the iterval -a, a, where a >? + 4 cos si We wat a - -a -a a f d =, so we ' re looig for a odd fuctio. The ol odd fuctio i the list is si. If = - d the approimatio is ad =, the whe uler' s method with a step size of.5 is used to approimate,.75.5.75.5 We start at the poit,, ad = + -.5 =.5, ad = + -.5.5 = - =.75,.75 D
4. Give the graph of = f show below, which of the followig values is the largest? - - f h - f f f - f - f - f - f =. This is the ol positive value i the list. The aswer is A 5. Which of the followig fuctios satisfies < f < f < f for all? f = e - f = e f = e f = e f = e The correct aswer is f = e, sice f = e ad f = 4 e ad f < f < f for all values of, B. Suppose L is the liearizatio formula for f = log at = 4. Which of the followig is true? L 4. overestimates f 4., ad L.9 overestimates f.9 L 4. uderestimates f 4., ad L.9 uderestimates f.9 L 4. overestimates f 4., ad L.9 uderestimates f.9 L 4. uderestimates f 4., ad L.9 overestimates f.9 Noe of the above are true - f = log so f = ad f = because the fuctio is cocave dow i l l the eighborhood aroud = 4, the liearizatio formula will be a overapproimatio o either side, A 7. h h + h e - t dt = - e - e e The it does ot eist h h + h e - t dt = h + h e - t dt h Now, assume that the fuctio e - t has a atiderivative, F t, F + h - F ad D ad this fits the defiitio of the derivative, so F = e - = e 8. D e cos d = -
e cos e cos - e - cos 5e cos e 5 cos + si The derivative of a costat is A 9. If g =, the g = The fuctio g ca be piecewise defied as : g = for so g = for > - for < - for < so g = B. If e si d =, the e si 4 - d = - - e si 4 - d u = 4 - ad du = - d or - du = d - e si u du or e si u du = =