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Math105N final exam

HEAL100 Health Care — exam and revision material · Other

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Health Care exam material, 7 pages. Originally published on StudyMerit.

Pages
7
Academic year
2025/2026
Language
English

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(1) Let f ( x ) = 1 − cos ( √ x ) x 3 − 4 x 2 . (a) (7 points) Find the domain of f . (b) (9 points) Check if f has any vertical asymptotes. Show all your work and find all relevant limits. Do not use L’Hˆ opital’s Rule. (c) (9 points) Check if f has any horizontal asymptotes. Show all your work and find all relevant limits. Do not use L’Hˆ opital’s Rule. messages.downloaded_by lOMoARcPSD|6879242

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(2) (a) (11 points) Let g be a differentiable function on the set of real numbers R . Suppose that g ′ is decreasing on R . Use the Mean Value Theorem to show that g ( x + 1) − g ( x ) < g ′ ( x ) < g ( x ) − g ( x − 1) , for every real number x . (b) Let f be a continuous function on R with f (1) = 3. Suppose that f ( f ( x )) + f ( x ) = 8 , for every real number x . (i) (8 points) Show that there is c ∈ (1 , 3) such that f ( c ) + c = 6. (ii) (11 points) Show that the graph of f is passing through the point P (3 . 01 , 4 . 99). messages.downloaded_by lOMoARcPSD|6879242

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(3) Let f ( x ) = 6 x 2 − 6cos x − 1 . (a) (6 points) Find f ′ ( 2 π 3 ) . (b) (8 points) Find all local extreme values (local maximum and local minimum values) of f on the interval ( − π 2 , π 2 ) , if any. (c) (7 points) Find all the intervals where f is concave up or concave down. Are there any inflection points for f ? Explain. messages.downloaded_by lOMoARcPSD|6879242

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