Skip to content
Solved Assignments

Advanced Calculus and Mathematical Analysis: Derivatives, Integrals, and Graphical Analysis

This solved assignment explores derivatives, integrals, and graphical analysis in advanced calculus.

Preview 3 of 7 pages

See the first 3 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    Advanced Calculus and Mathematical Analysis: Derivatives, Integrals, and Graphical Analysis 1) Find the slope of curve as ( ) ( ) ( ) ( ) ( ) 3 3 2 2 9 3 9 3 3 9 0 3 9 d h t t t dt d d d t t dt dt dt t t  = − − = − − = − − = − At 3 t = slope is ( ) ( ) 2 3 3 3 9 27 9 18 h  = − = − = Write the equation of line with slope 18 m = and point ( ) ( ) 1 1 , 3, 3 t y = − as ( ) ( ) ( ) 1 1 3 18 3 3 18 54 18 57 y y m t t y t y t y t − = − − − = − + = − = − Thus, the equation of tangent line is 18 57 y t = − . Hence, the correct option is B . 2) The volume of sphere at 3.0 r = is ( ) ( ) 3 1 3 4 3.0 3 36 3.14 113.04 cm V  = = = The volume of sphere at 3.1 r = is ( ) ( )( ) 3 2 3 3 4 3.1 3 4 3.14 3.1 3 124.72 cm V  = =  The change in volume is 3 3 2 1 3 124.72 cm 113.04 cm 11.68 cm V V − = − = Hence, the correct option is C . 3) As x tens to2 from left, the function value is ( ) 2 2 2 6 + = and as x tens to 2 from right, the function value is ( ) 2 4 6 + = , so to remove the discontinuity ( ) 2 f must be equal to 6. Hence, the correct option is B . 4) To find the velocity function, differentiate position vector with respect to t as ( ) ( ) 2 2 2 2 2 2 1 2 2 d v t t dt t t = + = + = + At 1 t = , ( ) 1 1 2 2 1 1 4 1 m/sec 2 v = +  = = Hence, the correct option is B . 5) Since the slope of line is positive in interval ( ) 5, 3 − − and ( ) 0,3 , so 0 f   in this interval. Since the slope of line is negative in interval ( ) 3,0 − , so 0 f   in this interval.

  2. Page 2
  3. Page 3
    Read page 3 as text

    Since the slope of line is constant in interval ( ) 3,6 , so 0 f  = in this interval. Hence the correct graph is C . 6) The derivative of 2 0.05 d v v = + with respect to v is ( ) ( ) ( ) ( ) 2 2 0.05 0.05 0.05 2 1 0.1 1 d d d v v v v dv dv dv v v + = + = + = + At 46 v = , ( ) 0.1 46 1 4.6 1 5.6 + = + = Hence the correct option is A . 7) Let the length of rectangle is x feet and width of rectangle is y feet. Since the area is 680 square feet, so 680 680 xy y x = = The cost is given by ( ) ( ) 7 2 6 2 14 12 680 14 12 8160 14 C x y x y x x x x = + = +   = +     = + Derivative of cost function is 2 8160 14 C x  = − Set C  equal to zero and solve for x 2 2 2 8160 14 0 8160 14 8160 14 8160 14 24.1 x x x x − = = = =  So, value of y is 680 24.14 28.2 y =  Hence the correct option is C . 8) Find the composite function as ( ) ( ) 7 2 7 2 7 2 7 7 x g f x g x x x −   =     −   = +     = − + = Hence, correct option is A . 9) Integrate the function as

4 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.