NAME: Calculus I - Spring 04 Midterm Exam I, Marc 5, 04 In all non-multiple coice problems you are required to sow all your work and provide te necessary explanations everywere to get full credit. In all multiple coice problems you don t ave to sow your work.
. Find te domain of f(x) x+ x+5. A (, 5] B (, ] C [ 5, ] Correct D [, 5] E [, 5] Solution: Te domain of x is all real numbers suc tat x 0, tat is (,]. Te domain of x+5 is all real numbers suc tat x+5 0, tat is [ 5, ). Terefore te domain of f(x) x+ x+5 is (,] [ 5, ) wic is [ 5,]. Info: Te average in te class for tis problem was 75.6%.. Te function f(x) sin x is A even Correct B odd C neiter even nor odd Solution: To make te solution clearer, let us write f(x) as (sinx). We ave f( x) (sin( x)) ( sinx) (sinx) sin x f(x) So, f( x) f(x), terefore f is an even function. Info: Te average in te class for tis problem was 58.5%.
3. Find ( +4x x). A 0 B C Correct D 3 ( +4x x) x +4x x ( x +4x x)( x +4x+x) ( +4x+x) ( x +4x) x +4x+x +4x x +4x+x 4x x +4x+x 4x +4x+x 4x x x +4x x + x 4 +4x + x x 4 + 4x 4 + + 4 x + 4 +0+ Info: Te average in te class for tis problem was 8.9%. 4 + 4 + 4 3
4. Find x x 3 8 4. A 0 B C D 3 Correct x 3 8 x 4 x 3 3 x (x )( +x+4) x (x )(x+) x +x+4 x+ + +4 + 4 3 Info: Te average in te class for tis problem was 5.4%. 4
5. Find x x+ 3. x A 9 Correct B 9 C 3 D 3 x x+ 3 x ( 3(x+) x+ ) 3 x 3(x+)(x ) 3(x+) x+ 3(x+) 3 x 3(x+)(x ) x 3 (x+) 3(x+)(x ) 3 x x 3(x+)(x ) x x 3(x+)(x ) (x ) x 3(x+)(x ) x 3(x+) 3(+) 9 Info: Te average in te class for tis problem was 8.7%. 5
6. Find x 0 sin3x 4x. A Does not exist B 3 4 Correct C 4 3 D 0 sin3x sin3x x 0 4x 3x 3x x 0 4x Info: Te average in te class for tis problem was 57.3%. [ ] sinu u 0 u 3x x 0 4x 3 x 0 4 3 4 7. Find x +x. A Does not exist and neiter nor B Correct C D Solution: Note tat +x 0 as x. Terefore [ ] x +x 0 Info: Te average in te class for tis problem was 5.%. 6
8. Find x +. A B C Does not exist and neiter nor D Correct x +x +( ) + Info: Te average in te class for tis problem was 59.8%. 9. Find x +x. A B C D Does not exist and neiter nor Correct Solution: By Problem 7, it follows tat So terefore x +x x +x On te oter and, since +x 0+ as x +, [ ] x + +x 0 + x +x x + +x does not exist and neiter nor. Info: Te average in te class for tis problem was 67.%. 7
0. Find x (+x). A Does not exist and neiter nor B C D Correct Solution: Since we square +x, it follows tat (+x) approaces 0 from te rigt as x. Terefore [ ] x (+x) 0 + Info: Te average in te class for tis problem was 75.6%.. If f(x) x, wic one of te following its correspond to f (a)? A B C D x 0 a+ a a+ a a a x a x a Correct E a 0 x a x a Solution: By te definition of te derivative, f (a) f(a+) f(a) (a+) a a+ a Info: Te average in te class for tis problem was 75.6%. 8
. Let f(x) be a function wic is differentiable for all x values. Wic of tese is te derivative of g(x) f( x)? A B f ( x) (f( x)) f ( x) x(f( x)) C f ( x) x(f( x)) D f ( x) x(f( x)) E f ( x) x(f( x)) Correct Solution: We can find g (x) in two different ways. Eiter [ ] g (x) f( [ (f( x)) ] ( )(f( x)) [f( x)] x) (f( x)) [f( x)] (f( x)) f ( x) ( x) (f( x)) f ( x) x or [ ] g (x) f( f( x) [f( x)] x) (f( x)) f ( x) x(f( x)) 0 f( x) f ( x) ( x) (f( x)) f ( x) x (f( x)) f ( x) x(f( x)) Info: Te average in te class for tis problem was 56.%. 9
3. Let f(x) cosx, ten te second derivative of f is A sinx B sinx C cosx D cosx Correct and f (x) (cosx) sinx f (x) (f (x)) ( sinx) (sinx) cosx Info: Te average in te class for tis problem was 90.%. 4. Let f(x) (+3x) 8, ten f (x) is A 4(+3x) 7 Correct B 8(+3x) 7 C 3(+3x) 7 D (+3x) 7 E 8(+3x) 9 f (x) [ (+3x) 8] 8(+3x) 8 (+3x) 8(+3x) 7 ( +(3x) ) 8(+3x) 7 ( +3(x) ) 8(+3x) 7 (0+3 ) 8(+3x) 7 3 4(+3x) 7 Info: Te average in te class for tis problem was 89%. 0
5. Suppose tat s(t) +5t t is te position function of a particle, were s is in meters and t is in seconds. Find te particle s instantaneous velocity at time t s. A m/s B 0 m/s C m/s D m/s E 3 m/s Correct Solution: Since te instantaneous velocity v(t) is s (t), we ave s (t) (+5t t ) +(5t) (t ) +5(t) (t ) 0+5 t 5 4t terefore te particle s instantaneous velocity at time t s is s () 5 4 5 8 3 m/s Info: Te average in te class for tis problem was 96.3%.
. Let f(x) x+ 3x. (a) Find all orizontal asymptotes of f. x+ 3x x+ 3 x+ x 3 x + x x 3 +0 3 3 0 3 + x and x x+ 3x x x+ 3 x x+ x 3 x x + x x 3 x 0 x 3 3 0 3 Terefore te function f(x) x+ as two orizontal asymptotes y and y. 3x 3 3 (b) Find all vertical asymptotes of f. Solution: Te vertical asymptotes of f are x 3 and x 3, since 3 0 and x+ 0 at x ± 3. Info: Te average in te class for tis problem was 55%.
. Let x+5 if x < f(x) sin(x ) if x x if x > Find all points were f is discontinuous. Use its to justify your answer! Solution: We first note tat x+5 is continuous everywere, since it is a polynomial sin( ) is continuous everywere, since it is a composition of a polynomial and sinx x is continuous at any point x >,since (, ) is in te domain of x Terefore te only potential points of discontinuities of f are x and x. Let us sow tat f is indeed discontinuous at tese points. We ave and Since x f(x) ( x+5) +5 4 x x +f(x) ) sin(( ) ) sin( ) sin0 0 x +sin(x x f(x) x +f(x) it follows tat f is discontinuous at x. Simiarly, and Since x f(x) ) sin(( ) ) sin( ) sin0 0 x sin(x it follows tat f is discontinuous at x. x +f(x) x x + x f(x) x +f(x) Info: Te average in te class for tis problem was 69.8%. 3
3. Let f(x) x+5. Use te definition of te derivative to find f (x). Solution : We ave f (a) x a f(x) f(a) x a x a ( x a x+5 a+5 x a x+5 a+5)( x+5+ a+5) (x a)( x+5+ a+5) ( x+5) ( a+5) x a (x a)( x+5+ a+5) (x+5) (a+5) x a (x a)( x+5+ a+5) x+5 a 5 x a (x a)( x+5+ a+5) x a x a (x a)( x+5+ a+5) (x a) x a (x a)( x+5+ a+5) x a x+5+ a+5 a+5+ a+5 a+5 a+5 4
Solution : We ave f (x) f(x+) f(x) ( (x+)+5 x+5 (x+)+5 x+5)( (x+)+5+ x+5) ( (x+)+5+ x+5) ( (x+)+5) ( x+5) ( (x+)+5+ x+5) ((x+)+5) (x+5) ( (x+)+5+ x+5) x++5 x 5 ( (x+)+5+ x+5) ( (x+)+5+ x+5) (x+)+5+ x+5 (x+0)+5+ x+5 x+5+ x+5 x+5 x+5 Info: Te average in te class for tis problem was 8.5%. 5