A piecewise function is defined for all real numbers by h(x)= cases a x^ 3 +b x^ 2 +c…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

A piecewise function

h

is defined for all real numbers by

h(x)=\begin{cases} a x^{3}+b x^{2}+c x+d, & x\le 1,\\ k\sin(\pi x)+1, & x>1, \end{cases}

where

a,\,b,\,c,\,d,

and

k

are constants. Additional information: •

h(0)=0

. •

h'(0)=2

. (1) Part 1: (a) Write the equations that must be satisfied to ensure that

h

is continuous at

x=1

. (2) Part 2: (b) Derive the equations that guarantee the first derivatives of the two pieces agree at

x=1

. (3) Part 3: (c) Derive the equations that guarantee the second derivatives of the two pieces agree at

x=1

. (4) Part 4: (d) Using the conditions from parts (a)–(c) together with the information

h(0)=0

and

h'(0)=2

, solve for the six unknown constants

a,\,b,\,c,\,d,

and

k

. Clearly show each algebraic step in your solution. (5) Part 5: (e) Verify explicitly that the function you obtained in part (d) has a continuous second derivative at

x=1

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