MAC FINAL EXAM REVIEW. Find each of the following its if it eists, a) ( 5). (7) b). c). ( 5 ) d). () (/) e) (/) f) (-) sin g) () h) 5 5 5. DNE i) (/) j) (-/) 7 8 k) m) ( ) (9) l) n) sin sin( ) 7 o) DNE p) () q). ( ) r) Find ( ). ( ) s) 8. (-/9) t) Find. 8 65 sint sin u) (/5) v) t sin5t w) f ( ) Where (9) ( ) (/) if f ( ) DNE 9 if f ( ). If f ( ) and g( ), find. (-/) c c c g ( ). Use definition of the it to show that ( ), find. ( ) (9)
9 if f if f is continuous at, k if, what is the value of k? (6). A function f is defined by 5. If a ball is thrown into the air with a velocity of ft. /s, its height (in feet) after t seconds is given by s( t) t 6t. Find the velocity when t =. ( ft. /s) 6. Find an equation of the tangent line to the curve at the given point y e, (,) (y = ) 7. Use a linear approimation L() to an appropriate function f (), with an appropriate value of a, to estimate 79 5. ( 5. 959) 7 8. Differentiate: 5 5 a) f ( ) e cos f ( ) e (sin 5cos) dy b) ( y ) f ( ) f ( ) f ( ) d ( ) c) f( ) ( ) d) y tan ( e e ) y e e) cos y -csc(y) f) y g) y (7 ) h i) y cot cot csc y y (ln ) j) y sin cos sin () k) y ln cos tan( ) l) sin ( )cos( ) m) y 5sin ( ). (6 )cos( )sin ( )
9. a) Find all vertical asymptotes of f ( ). 6 (-, ) b) Find all horizontal asymptotes of g ( ). 6 (/). Determine the value of c for which f () is continuous for all real numbers when, 5 f ( ). c, 5 (6/5). Find, if y 6 5 6 5 5. Consider the relation y. Find the slope of the tangent line at the point (, -) (). Determine the value of c that satisfies the Mean Value Theorem for derivatives for interval [, ]. ( c ) y on the d y. Calculate d for y ( ). 5. Find an equation for the tangent line to the graph of f ( ) at the point where =. y = 5 6. The position function for the movement of a particle is given by s ( t ) where s is measured in feet and t is measured in seconds. Find the acceleration of the particle at second. () 7. Find the values of for all points on the graph of f ( ) 5 6 at which the slope of the tangent line is. (/, ) 8. As a balloon in the shape of a sphere is being blown up, the volume increases at a rate of cubic inches per second. At what rate is the radius increasing when the radius is inch? v ( / ) r ( ) 9. An object projected vertically from a height of 8 feet with an initial velocity of 6 ft. / sec has a height, h (t), given by h(t) 6t 6t 8 where t is time in seconds. Find the velocity at which the object hits the ground. (-96 ft. / sec). Find the differential dy. a) If y. d 6 y d b) If y = sec (). dy = sec( ) tan( ) d c) y ( 5),, d. 5 dy 6( 5) d =.8. Determine the value of c that satisfies the Mean Value Theorem for derivatives for
f on,.. Determine intervals where f ( ) is increasing or decreasing. Dec (-, ) U (, ). Find all critical numbers for the function f ( ). -/, -/ 5. Find all etrema in the interval [, ] if y sin cos. (, ),(, ) 5. Let R be the region bounded by y 6, =, =, and y =. Find the area of the region. (9) 6. Find the area determined by the two curves and (/6) 7. A rectangular bo is to be made by cutting out equal squares from each corner of a piece of cardboard inches by 6 inches and then folding up the sides. What must be the length of the side of the square cut out if the volume is to be maimized? What is the maimum volume? square cut-out: inches by inches; volume: in 8. If f ( ) d, f ( ) d, g( ) d and g ( ) d, use the properties of definite integrals to calculate each of the following integrals. (a) [ f ( ) g( )] d (b) [ f ( ) g( )] d (c) [ f ( ) 5g( )] d - (d) [ f ( ) g( )] d (e) [ f ( ) g( ) ] d 5 9. Find all intervals for which the graph of y 8 is concave downward. (-, ) U (, ). Let f ( ). Use the second derivative test to find all relative etrema. Ma @ =, Min @ = / 5. Given the function defined by f ( ), find all values of for which the graph of f is concave up. (, ) (, ). Determine whether Rolle s Theorem can be applied to the function, f ( ) ( )( ) on the interval [, ]. If it can, then find all values of c guaranteed by the theorem. -, /. The sum of two numbers is. Find the maimum of the product of the two numbers. () 6. Let f ( ).
(a) Find the intervals on which the graph is increasing and decreasing. (b) Find any local etrema. (c) Find the intervals on which the graph is concave up and concave down. (d) Find any inflection points. (e) Find all asymptotes. (a) Dec: (, ), (, ) (b) No local etrema (c) Concave upward: (, ); concave downward: (, ) (d) No inflection points (e) Horizontal asymptote: y = ; vertical asymptote: = 5. Find all points of inflection for the graph of the function f ( ) ( ). (, -), (, ) 6. The sum of two numbers is. Find the two numbers that would give a maimum product. 6, 6 7. Find the maimum value and minimum value of / f ( ) ( ) on [, ]. Ma: f ( ) 9 ; min: f () = 8. Find the intervals for which the function If y Find the differential dy. 9. f ( ) is increasing or decreasing. Dec (-, ) U (, ), Inc. (, ) 8 ( ) d. A spherical balloon is being blown up at a rate of cm /min. At what rate is its radius r 5 changing when r is cm?. 97 cm/min 6. Determine if the Mean Value Theorem (for derivatives) applies to f ( ) sin() on the interval,. If it does, find the value of c whose eistence is guaranteed by the Mean Value Theorem. (Use the inverse trigonometric functions on your calculator to approimate the value(s) for c.) MVT applies since f () is continuous on, and differentiable on,. C.. Find an equation of the tangent line to the graph of the equation 9y y 6 at the 9 point (, 6). y 6. Find the average value function of f sin over the interval [, / ] ( ). Find the area determined by the curves f ( ) 5. Evaluate each of the following integrals and g( ) / 9 a. d ln b. cos d
C. dt 7/8 d. t ( ) d e. 8 d C f. sin (sin ) cos d C 6 C g. / 6 sin cos d 6 5 h) cos ( 5) sin( 5) d i) 6 9 d j) d k) d. 5 l) d. sec tan m) d. tan 6 cos ( 5) C 8 ln( ) c ( ) / C c 5 7 5 7 +c sec( ) c n) / cos d? ln () sin o) ( ) d. p) sin ()cos() d. ( ) 5 c sin ( ) c q) csc d. cot( ) r) d. () s) d. ( ) ( ) 5
t) t t 9 dt u) g( ) t dt v) d g ( ) C w) g ( ) ( t sint dt g ( ) ( )[( ) sin( )] ) / ) ( cos ) d / 6. Estimate the area under the graph of f ( ) from = to = 5 using four rectangles and right endpoints. Repeat using left endpoints. R 77 / 6 L 5/ 7. If a snowball melts so that its surface area decreases at a rate of sq. cm/min, find the rate at which the diameter decreases when the diameter is cm. (Note: S r ) Decreases at a rate of cm/ min 8. For each of the following functions: Find the intervals where it is increasing and decreasing. Find the local maimum and minimum values. Find the intervals of concavity and the inflection points. a) f ( ) = vertical asymptote ( ) (-, -¼) local minimum CU: (-, ) and (, ) Inflection point (-, -/9 5 b) f ( ) 5 local maimum: (-, 5) local minimum: (, ) Inflection points:, 9. Find the average value of the given function on the indicated interval. (a) f ( ) 8 ; [, 9] -9 688 (b) g( t) 8 t; [, 6] (c) f ( ) sin( ); [, ] 7 8 5. Find the area of the region enclosed by the curves y, y 8/ 5. If f ( ) cos[cos( )], find f (.). f (.). 79