f(x) p(x) =p(b)... d. A function can have two different horizontal asymptotes...

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1 Math Final Eam, Fall. ( ts.) Mark each statement as either true [T] or false [F]. f() a. If lim f() =and lim g() =, then lim does not eist !5!5!5 g() b. If is a olynomial, then lim!b () =(b).... c. If f is continuous at c, then f is differentiable at c.... d. A function can have two different horizontal asymtotes e. If f has a local minimum at c, then f (c) =.... f. If f (c) =, then f has an inflection oint at c.... g. If f and g are both continuous functions on the interval [a, b], then Z b Z b Z b f()g() = f() g().... a a a h. If d (f()) = d (g()), then f() =g().... i. If f is differentiable and f() >, then d f() = f () f().... j. If f () eists and is nonzero for all, then f() 6= f()..... (8 ts.) An engineer needs to build a rectangular bo that has a square base and no to with an outside surface area of square feet. What are the dimensions of the bo meeting these restrictions that has the greatest ossible volume? (For full credit, you must indicate what function you are maimizing and on what interval.)

2 Math Final Eam Page of 7 Points: /. Find dy for each of the following. (You don t need to simlify your answers.) (a) ( ts.) y = e +sin + (b) (5 ts.) y = + arctan (c) (6 ts.) y = +sec e tan (d) ( ts.) y =(+sin) +cos (e) ( ts.) y =(ln(9 +5)). (6 ts.) Find the equation of the tangent line to the grah y = at the oint where =. 5. (7 ts.) Find dy along the curve defined by +y siny =5. 6. (8 ts.) (a) Find the linearization L() of f() = + at a =. (b) Use your answer to art (a) to aroimate.

3 Math Final Eam Page of 7 Points: / 9 7. ( ts.) Using the aes below, sketch the grah of a function f() with all the following roerties: lim! f() = and lim f() =+.! + f() =. lim!+ f () does not eist. f() = and lim! f () > on (, ) and (, ), while f () < on (, ), (, ) and (, ). f () > on (, 6) and (, ), while f () < on ( 6, ) and (, ). y Using the following table of values for f,g,f and g, answer the questions below. (a) ( ts.) If F () =f()g(), find F (). f() g() f () g () / (b) ( ts.) If G() =f(g()), find G ().

4 Math Final Eam Page of 7 Points: / 9. (5 ts.) Given that the function f() is as in the grah below, and that g() = determine the following five values: Z f(t), g() = g(5) = g( ) = g () = g (6) = y f() Suose the derivative of f() is f () = and the domain of f is 6=. (a) ( ts.) Find all critical numbers for f. (b) ( ts.) Find all intervals where f is increasing, and all intervals where f is decreasing. (c) ( ts.) For each critical number, determine if it gives a local maimum, local minimum, or neither. (d) (6 ts.) Find all intervals where f is concave u, and all intervals where f is concave down. (e) ( t.) Find the -coordinates of any inflection oints.

5 Math Final Eam Page 5 of 7 Points: / 8. Find the following limits, find the infinite limit, or state that the limit does not eist. 6 (a) ( ts.) lim!6 6 (b) ( ts.) + lim! (c) (5 ts.) lim! cos() (d) (7 ts.) lim! + + (e) (5 ts.) lim tan!+ 5. (7 ts.) Suose that f() = and f () ale for all. How large can f() ossibly be?. (7 ts.) Use the definition of derivative to find f () if f() =. (Use only algebraic techniques to evaluate the limit; do not use L Hôital s Rule.). (9 ts.) Find the absolute maimum and minimum values for f() =cos sin on the interval [, ].

6 Math Final Eam Page 6 of 7 Points: / 6 5. (8 ts.) Suose we know that Z f() =5and Use this information to evaluate the following: (a) Z f() Z f() =. (b) Z ( f()) 6. (5 ts.) Use a Riemann sum to aroimate Z ln. Use n =equal-length subintervals and take the samle oints to be the right-hand endoints. (Do not simlify your answer.) 7. An elevator is moving u and down in a vertical shaft, and we measure its height (in meters) above ground level as a function of time (in minutes). Suose the elevator s acceleration is a(t) =t 6. (a) ( ts.) Assume that the elevator is traveling uward at 8 meters er minute at time t =. What is its velocity minutes later? (b) ( ts.) Assume that the elevator is meters above ground level at t =. What is its height minutes later? (c) (5 ts.) What is the total distance travelled by the elevator during the time interval ale t ale?

7 Math Final Eam Page 7 of 7 Points: / 5 8. Find the following indefinite integrals: Z (a) (7 ts.) 5 / +sectan + (b) (7 ts.) Z 5 (c) (7 ts.) Z cos sin 7 9. Evaluate the following definite integrals (simlify your answers): (a) (5 ts.) Z / sin (b) (9 ts.) Z +

8 Math Final Eam, Fall, with Solutions. ( ts.) Mark each statement as either true [T] or false [F]. f() a. If lim f() =and lim g() =, then lim does not eist F!5!5!5 g() b. If is a olynomial, then lim!b () =(b).... T c. If f is continuous at c, then f is differentiable at c.... F d. A function can have two different horizontal asymtotes T e. If f has a local minimum at c, then f (c) =.... F f. If f (c) =, then f has an inflection oint at c.... F g. If f and g are both continuous functions on the interval [a, b], then Z b Z b Z b f()g() = f() g().... F a a a h. If d (f()) = d (g()), then f() =g().... F i. If f is differentiable and f() >, then d f() = f () f().... T j. If f () eists and is nonzero for all, then f() 6= f().... T. (8 ts.) An engineer needs to build a rectangular bo that has a square base and no to with an outside surface area of square feet. What are the dimensions of the bo meeting these restrictions that has the greatest ossible volume? (For full credit, you must indicate what function you are maimizing and on what interval.) Calling the base width and the height y, the volume is V = y and the surface area is S = +y. S =so = +y and solving for y gives y =( )/(). Substituting into the volume formula to eliminate y and roduce a function of one variable, V () = ( )/() =( )/ The hysical constraints are y, y =( )/() > and thus ale. Differentiating V () gives V () =( =. )/ with only one critical number in the interval: The values of the function at the endoints and at the critical oint inside the interval are V () =, V () = and V ( ) =, so the maimum volume is, for base width =feet and thus height y =feet.

9 Math Final Eam, with Solutions Page of. Find dy for each of the following. (You don t need to simlify your answers.) (a) ( ts.) y = e +sin + y =e +cos, (b) (5 ts.) y = + arctan y = / + (ln ) + (c) (6 ts.) y = +sec e tan y = (e tan)( sec tan ) ( + sec )(e sec ) (e tan) (d) ( ts.) y =(+sin) +cos y =cos +cos +(+sin) +cos ( sin) (e) ( ts.) y =(ln(9 +5)) y =ln(9 +5) 9 (8 +5) (6 ts.) Find the equation of the tangent line to the grah y = at the oint where =. y () = + +5, y ( ) = +5=, y( ) = + 5+7= tangent line equation is y =( +) or y =( +)+= +7

10 Math Final Eam, with Solutions Page of 5. (7 ts.) Find dy along the curve defined by +y siny =5. Imlicit diff. gives +(y + dy ) Gathering terms on each side: ( Solving for the derivative: dy dy cosy =. cosy) dy = y = y cosy 6. (8 ts.) (a) Find the linearization L() of f() = + at a =. f () = + = + f () = = + 5 ; f() = + = 5 y 5= ( ) 5 L() =y = (b) Use your answer to art (a) to aroimate. () = 5

11 Math Final Eam, with Solutions Page of 7. ( ts.) Using the aes below, sketch the grah of a function f() with all the following roerties: lim! f() = and lim f() =+.! + lim f() = and lim f() =.!+! f () does not eist. f () > on (, ) and (, ), while f () < on (, ), (, ) and (, ). f () > on (, 6) and (, ), while f () < on ( 6, ) and (, ). lim! f() = and lim f() =+.! + lim f() = and lim f() =.!+! f () does not eist. f () > on (, ) and (, ), while f () < on (, ), (, ) and (, ). f () > on (, 6) and (, ), while f () < on ( 6, ) and (, ). y

12 Math Final Eam, with Solutions Page 5 of 8. Using the following table of values for f,g,f and g, answer the questions below. (a) ( ts.) If F () =f()g(), find F (). f() g() f () g () / F () =f ()g()+f()g (), so F () = f ()g() + f()g () = 5 7+( ) 9=. (b) ( ts.) If G() =f(g()), find G (). G () =f (g())g (), so G () = f (g())g () = f (7) 9= 9=9. 9. (5 ts.) Given that the function f() is as in the grah below, and that g() = determine the following five values: Z f(t), g() =.5 g(5) = g( ) = g () = g (6) = y f() Suose the derivative of f() is f () = and the domain of f is 6=. (a) ( ts.) Find all critical numbers for f. The derivative f is defined on the whole domain, so critical oints can only occur where it vanishes, which is where its numerator vanishes: the critical numbers are ±.

13 Math Final Eam, with Solutions Page 6 of (b) ( ts.) Find all intervals where f is increasing, and all intervals where f is decreasing. f can change sign only at the critical numbers and at =where it is not defined. The sign chart (, ) (, ) (, ) (, ) f () + + shows that f is increasing on the intervals (, ) and (, ); decreasing on the intervals (, ) and (, ). (c) ( ts.) For each critical number, determine if it gives a local maimum, local minimum, or neither. The increasing/decreasing behavior seen above shows (by the first derivative test) that there are local minima of f at each of = and =. (d) (6 ts.) Find all intervals where f is concave u, and all intervals where f is concave down. f () = d = d = +9 = ( +9)= ( )( +) This factorizes as f () = ( )( +), vanishes only at the roots ±, and can only change sign at those roots, so the sign table is (, ) (, ) [ (, ) (, ) f () + Thus, f is concave u on the two intervals (, ) and (, ) and concave down on the two intervals (, ) and (, ). (e) ( t.) Find the -coordinates of any inflection oints. The concavity changes at each of the oints = ± where f () =and nowhere else, so those are the inflection oints.. Find the following limits, find the infinite limit, or state that the limit does not eist. 6 (a) ( ts.) lim!6 6

14 Math Final Eam, with Solutions Page 7 of 6 lim!6 6 =lim!6 (b) ( ts.) + lim! ( 6)( +6) ( 6) =lim!6 ( +6) = = 6 = + Version : lim! + + Version : 6 = lim! + ( + ) +( + ) ( + ) +( + ) 6 = lim! = + lim! = lim +! + ( +)( ) = lim ositive! + (going to + )(negative) = (c) (5 ts.) lim! cos() lim! cos() (d) (7 ts.) = = lim! lim! lim! + + s = Alternate route: = = lim! (e) (5 ts.) q + + = Ĥ =lim! sin() ( +) = lim +! r ++ lim! lim tan!+ Ĥ =lim! cos() = 9cos() = 9 = + + = lim q+! + + = 5 s q + lim tan!+ tan 5 = lim!+ 5 / 5 sec Ĥ = lim!+ ( 5/ ) / =5 lim!+ sec 5 =5sec () = 5 =5

15 Math Final Eam, with Solutions Page 8 of. (7 ts.) Suose that f() = and f () ale for all. How large can f() ossibly be? Using the MVT with a =, b = f(b) f(a) = f (c) ale, so f(b) f(a) ale (b a), b a and a =, b =gives f() f() ale ( ). Thus f() ale f() +,, and as f() =, f() ale +ale : the largest that f() can be is.. (7 ts.) Use the definition of derivative to find f () if f() =. (Use only algebraic techniques to evaluate the limit; do not use L Hôital s Rule.) f f( + h) f() () =lim h! h =lim h! h + h =lim h! h =lim h! h =lim h! h =lim h! = + h + h + h + h + h =lim + + h + + h h! ( + h) =lim + h( + + h) h! h + h( + + h) = ( + ) / + h h + h h + h( + + h). (9 ts.) Find the absolute maimum and minimum values for f() =cos sin on the interval [, ]. f () = sin cos f () = ) = / f() = cos() sin() = = f() = cos( ) sin( ) = = f( /) = cos( /) sin( /) = Maimum is ; minimum is =

16 Math Final Eam, with Solutions Page 9 of 5. (8 ts.) Suose we know that Z f() =5and Use this information to evaluate the following: (a) (b) Z Z f() = - ( f()) = 6. (5 ts.) Use a Riemann sum to aroimate Z Z f() =. ln. Use n =equal-length subintervals and take the samle oints to be the right-hand endoints. (Do not simlify your answer.) Evaluate ln... at =, 6, 8,,... add these,... and multily by. R =(ln+ln6+ln8+ln()). 7. An elevator is moving u and down in a vertical shaft, and we measure its height (in meters) above ground level as a function of time (in minutes). Suose the elevator s acceleration is a(t) =t 6. (a) ( ts.) Assume that the elevator is traveling uward at 8 meters er minute at time t =. What is its velocity minutes later? v = t 6t + C v() = 8 gives C =8, so v = t 6t +8 At t =, v = 6 +8= (b) ( ts.) Assume that the elevator is meters above ground level at t =. What is its height minutes later? s = t t +8t + C s() = gives C =and so s = t t +8t + At t =, s = +8 +=6 (c) (5 ts.) What is the total distance travelled by the elevator during the time interval ale t ale? v = at t = (Also at t =, but unnecessary.) Distance = s() s() + s() s() = =6 + =7 or. 8. Find the following indefinite integrals: Z (a) (7 ts.) 5 / +sectan Z 5 / +sec tan + = 5/ + sec tan + C +

17 Math Final Eam, with Solutions Page of (b) (7 ts.) Z 5 Z 5 Z = 5 = 5 + C (c) (7 ts.) Z cos sin 7 u =sin() 7, du =cos() Z integral = du =ln u + C =ln sin() u 7 + C 9. Evaluate the following definite integrals (simlify your answers): Z + = + = ( + ) ( + ) = 7 (a) (5 ts.) Z / sin Z / sin = cos / = (cos / cos ) = = (b) (9 ts.) Z + u =+, du = Z Z 9 + = udu = 9 u/ 9 = 9 9/ = 9 Alternate solution: u Z =+, du = Z + = udu = 9 u/ + C = 9 + / + C Z + = 9 + / = 9 9/ = 9 6 = = 5 9