204k views
2 votes
The graph of the continuous function g, the derivative of the function f, is shown above. The function g is piecewise linear for -5 ≤ x < 3 and g(x) = 2(x-4)^2 for 3 ≤ x ≤ 6.

(a) If f(1) = 3, what is the value of f(-5) ?

(b) Evaluate
\int_(0)^(6) g(x)\; dx .

(c) For -5 < x < 6, on what open intervals, if any, is the graph of f both increasing and concave up? Give a reason for your answer.

(d) Find the x-coordinate of each point of inflection of the graph of f. Give a reason for your answer.

(This was on the no-calculator section of the recently-released AP Calculus AB 2018 exam so I appreciate it if you tried to limit calculator usage)

The graph of the continuous function g, the derivative of the function f, is shown-example-1

1 Answer

4 votes
a)
g=f' is continuous, so
f is also continuous. This means if we were to integrate
g, the same constant of integration would apply across its entire domain. Over
0<x<1, we have
g(x)=2x. This means that



f_(0<x<1)(x)=\displaystyle\int2x\,\mathrm dx=x^2+C


For
f to be continuous, we need the limit as
x\to1^- to match
f(1)=3. This means we must have



\displaystyle\lim_(x\to1)x^2+C=1+C=3\implies C=2


Now, over
x<-2, we have
g(x)=-3, so
f_(x<-2)(x)=-3x+2, which means
f(-5)=17.


b) Integrating over [1, 3] is easy; it's just the area of a 2x2 square. So,



\displaystyle\int_1^6g(x)=4+\int_3^62(x-4)^2\,\mathrm dx=4+6=10


c)
f is increasing when
f'=g>0, and concave upward when
f''=g'>0, i.e. when
g is also increasing.

We have
g>0 over the intervals
0<x<4 and
x>4. We can additionally see that
g'>0 only on
0<x<1 and
x>4.


d) Inflection points occur when
f''=g'=0, and at such a point, to either side the sign of the second derivative
f''=g' changes. We see this happening at
x=4, for which
g'=0, and to the left of
x=4 we have
g decreasing, then increasing along the other side.


We also have
g'=0 along the interval
-1<x<0, but even if we were to allow an entire interval as a "site of inflection", we can see that
g'>0 to either side, so concavity would not change.
User Iamjoosy
by
7.4k points