Question

Help please! (:
Integral problem
attached below..

Answer 1

Just find g' and g'' and evaluate at those values

Answer 2

for a just evaluate the integral at x=2

Answer 3

how do i find g' and g" if there's no function f(x) there's only a graph?

Answer 4

or do I use the formula of a semi circle?

Answer 5

The function they give you, g(x), is what I like to call an "area accumulator."
This is so because the upper bound of the integral is a variable, x.
It accumulates area starting from 0 and ends at x, or wherever you tell it to end.

(a) says to find g(2). So, we plug in x=2 into g(x).
If we look at our function, it says to integrate f(t) from t=0 to t=2. On this interval, the graph is half a semi-circle, or a quarter of a circle. The radius of this circle is 2.

The area of a circle is A = pi * r^2. So A = 4pi of a circle of radius 2. However, we want only a quarter of this, so we'll divide by 4.

So, g(2) = 4pi/4 = pi.

(b) Wants dg/dx at x=-1. The derivative, dg/dx tells us how fast (or slow) g(x) is changing. We can use the fundamental theorem of calculus to take the derivative of g(x).

d g(x) / dx = d/dx ∫ f(t) dt {from 0 to x}.

The d/dx cancels the integral, so we're left dg/dx = f(x).

At x=-1, f(-1) = 1. So dg/dx is equal to -1.

(c) This wants the rate of change OF the rate of change. The first derivative was just the graph f. The rate of change of the graph at x=2 is 0 because the tangent line at x=2 is horizontal.

So d^2 g / dx^2 at x=2 is 0.

Answer 6

OMG! Thank you! (: @TurtleMuffin

Answer 7

That made it sooo easy to follow and understand!!! Could you possibly help me with another problem? @TurtleMuffin I attached it!

Answer 8

Maxima and minima of a function occur when the derivative is zero, or in other words the slope of the function is zero.

F(x) is also an area accumulator here, because the upper bound of the integral is a variable. You plug in an 'x' and tell it how much area to accumulate.

If we derive F(x), we find that F'(x) = f(x) by the fundamental theorem of calculus like in the last problem.

Looking at the graph, f(x) is equal to 0 at x=1, x=3, x=5, x=7, and x=9. All of those are choices for maxima and minima.

Now we need to classify them further. A point is a maximum when the function increases, stops momentarily, and decreases, right? We could rephrase that and say when the function's slope is positive, zero, and then negative.

But function's slope = function's derivative!

F'(x) = f(x).

f(x) is positive from 0 to 1, and then negative from 1 to 3. So x=1 must be where F(x) has a maximum.

Similarly, f(x) is negative from 1 to 3, and then positive from 3 to 5. So, this says that F(x) was decreasing, stopped, and then starting increasing. This must mean that F(x) has a minimum at x=3.

The others can be classified the same way.

Answer 9

Part (c) wants to know if F(x) is increasing. I'm not too sure if this is asking for which intervals F(x) is increasing, or if it's asking if it's a monotonically increasing function.

It's not increasing everywhere, since it has minima. The work done for (a) and (b) find when it is increasing though.

(d)
F(x) is concave up when it's second derivative is positive. But the first derivative of F(x) is f(x).
So we have to approach this graphically now. We want to know when F''(x) = f'(x) > 0.

But derivative = slope! So when is the slope of f(x) increasing? Looks like it's from 2 to 4, and from 6 to 8. So this means F(x) is concave up on those intervals.

Answer 10

(e)
I'm not sure how to justify this part since F(2) by the looks of it is greater than 0. However, it's not guaranteed.

This is not an AP Calc question, right? It seems like it was written in haste.

(f)
F'(8) = f(8) and f(8) appears to be below 0.3, so yes.

(g)
F''(5) is f'(5). One can approximate it using a secant line. By the looks of it, it's probably 1.

Answer 11

It could be from AP Calc. but this is from my Calc 1 class. But for part (d) we can determine the concave up using the 2nd derivative of f(x) right?

Answer 12

The question asks to find when, if ever, is F(x) concave up. So we need to use the second derivative of F(x).

But the first derivative of F(x), is f(x), yeah?

So the second derivative of F(x) has to to be the derivative of f(x).

But the derivative of f(x) tells us how fast f(x) is changing, in other words the slope of f(x).

Answer 13

Symbolically:
F'(x) = f(x)
F''(x) = f'(x)

Answer 14

Ohhhh! okay! thank youu! (: