( ) = 1 t + t. ( ) = 1 cos x + x ( sin x). Evaluate y. MTH 111 Test 1 Spring Name Calculus I

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1 MTH Test Spring 209 Name Calculus I Justify all answers by showing your work or by proviing a coherent eplanation. Please circle your answers.. 4 z z + 6 z 3 ez 2 = 4 z z2 2ez Rewrite as 4 z + 6 z3 ez 2 an take the erivative term by term. 2. Given the function f (t) = t t t + tant, fin the instantaneous rate of change in f (t) at t =. f (t) = 2 3 t t + sec 2 t Rewrite as 3 t2 + t t + tant an take the erivative term by term. The term t t is the prouct of two functions so use the pprouct rule: t t ( ) = t + t 2 t. Evaluate f () = Fin the slope of the tangent line to the curve y = cos at = 6. y = cos sin Use the prouct rule cos ( ) = cos + ( sin ). Evaluate y 6 = ln = ln 2 Rewrite as ln an use the prouct rule: ln = 2 ln +. You coul also have use the quootient rule: ln = ln. 2

2 5. The vertical isplacement y (in cm) of the en of an inustrial robot arm for each cycle is 3 2 y = 2t 0 sect where t is the time (in sec). Fin the instantaneous vertical velocity at t = 4 sec. y = 3 t sect tan t Take the erivative term by term an evaluate 0 y 4 = Fin ( 3 sinh cosh ). 3cosh 2 + 3sinh 2 Hol out the 3 an use the prouct rule: 3 sinh cosh ( ) = 3( cosh cosh + sinh sinh ) 7. Fin the slope of the tangent line to f () = 3 ln at =. (Remember ln n = nln ) f () = ln Use the hint an rewrite as 3 2 ln. 3 2 ln ( ) = 2 32 ln + 3 = 2 3 ln. Now evaluate f () = 2 8. When woul the tangent line(s) to the curve y = e sin be parallel to the -ais? ( Restrict your finings the interval [, ] ) ( ) Use the prouct rule: e sin y = e sin + cos e ( sin + cos ) = 0 = 4, 3 4 ( ) = e sin + e cos. Now solve 9. ( 6 + ( 4 ) cot ) = 2 cot 4 csc 2 = 4 Take the erivative term by term. The erivative of the first term is 0 an use the prouct rule on the secon term. (( 4 ) cot ) 4 = 2 cot + ( 4 ) csc ( ). Evaluate at = 4 obtaining

3 0. Given f () = e, fin f ( 0). f () = e + e, f () = 2e + e To obtain f () use the prouct rule: e erivative of f ( ) term by term: e + e ( ) = e + e. To obtain f ( ) take the ( ) = e + ( e + e ). Evaluate f 0 ( ) = 2. Fin the angle of inclination, θ, to f () = 3e 2 at = 0. The angle of inclination is the arctangent of the slope of the tangent line! f () = 3e 2 ln 2 Take the erivative term by term : 3e 2 ( ) = 3 ln 2 θ = arctan( 3 ln 2) 49.3 f 0 ( ) = 3e 2 ln2 an evaluate 2. Fin the slope of the tangent line to f () = tan + tan at = 4 f () = sec 2 + sec 2 + tan Take the erivative term by term: ( tan + tan ) = sec 2 + ( tan + sec 2 ) Evaluate f ( 4 ) = 3 + 2

4 MTH Test Spring 209 Name Calculus I Justify all answers by showing your work or by proviing a coherent eplanation. Please circle your answers.. z 4 z z3 6 z2 = 4 z 2 2 z 2z Rewrite as 4 z 6 z 3 z 2 an take the erivative term by term. 2. Given the function f (t) = t 3 + t t + cost, fin the instantaneous rate of change in f (t) at t =. f (t) = 3 2 t + 3t 2 sint Take the erivative term by term. The secon term is a prouct of two functions so you must use the prouct rule: ( t t ) = t + t 2 t. Now evaluate f () = Fin the slope of the tangent line to the curve y = sin at = 6. y = cos + sin Evaluate y 6 = ln e 3 = e 3 Rewrite as ( e ) 3 ln 5. The vertical isplacement y (in cm) of the en of an inustrial robot arm for each cycle is 3 2 y = 2t tant 0 where t is the time (in sec). Fin the instantaneous vertical velocity at t = 4 sec. y = 3 t 0 sec2 t. Evaluate y 4 = 3 2 5

5 6. Fin ( 8 tanh coth). Hol out the 8 an use the prouct rule: 8( sech 2 coth tanh csch 2 ) = 0 7. Fin the slope of the tangent line to f () = 2 ln 2 at =. (Remember ln n = nln ) f () = 4 ln + 2 Use the hint an rewrite as 2 ( 2 ln ). Use the prouct rule : ( 2 ( 2 ln ) ) = 2 2 ln ( ) = 2 2 ln + 2 an evaluate f () = Fin the equation of the tangent line to f () = tan + 2cos at =. f () = sec 2 2sin Take the erivative term by term an evaluate f () = to get the slope ie. in y = m + b, m =. We now nee a point, lets try = 0 in the original function. f (0) = 2, we now have b = 2. Our final line is y = ( csc ) = csc 2 cot csc Take the erivative term by term. = 4 The erivative of the first term is 0 an use the prouct rule on the secon term: ( 4 csc ) 4 = csc + 4 ( csc cot ) 2. Evaluate at = we obtain Given f () = cos, fin f ( ). f () = cos sin ; f () = cos 2sin Use the prouct rule to obtain f () : ( cos ) = cos + ( sin ). Now take the erivative of f () : ( cos sin ) = sin ( sin + cos). Finally evaluate f ( ) =.

6 . Fin the angle of inclination, θ, to f () = 7 + 4e at = 0. The angle of inclination is the arctangent of the slope of the tangent line! f () = 4e + 7 ln 7 Take the erivative term by term: 7 + 4e ( ) 54.5 θ = arctan ln ( ) = 7 ln 7 + 4e an evaluate f (0) = ln When woul the tangent line(s) to the curve y = e sin be parallel to the -ais? ( Restrict your finings the interval [, ] ) ( ) y = e sin + cos Use the prouct rule: ( e sin ) = e sin + e cos. Now solve e ( sin + cos ) = 0 = 4, 3 4