X 0 0.5 1.0 1.5 2.0 f(x) 3 3 5 8 13 12.) A table of values...

x 0 0.5 1.0 1.5 2.0 f(x) 3 3 5 8 13 12.) A table of values for a continuous function f is shown above. If four equal subintervals of [0,2] are used, which of the following is the trapezoidal approximation of ??² f(x)dx? (A) 8 (B) 12 (C) 16 (D) 24 (E) 32 13.) What are all values of x for which the function f defined by f(x) = x³ + 3x² - 9x + 7 is increasing? (A) -3 < x < 1 (B) -1 < x < 1 (C) x < -3 or x > 1 (D) x < -1 or x > 3 (E) All real numbers 14.) Let f and g be differentiable functions with the following properties: (i) g(x) > 0 for all x (ii) f(0) = 1 If h(x) = f(x) * g(x) and h'(x) = f(x) * g'(x), then f(x) = (A) f'(x) (B) g(x) (C) e? (D) 0 (E) 1

Transcript

00:01 Consider the given table of values below.

00:02 And here we want to determine the trapezoidal approximation for the integral from 0 to 2 of f of x dx for n equal to 4.

00:14 Now the trapezoidal rule states that the integral from a to b of f of x dx, this is approximately delta x over 2 times f of x of 0 plus 2 f of x of 1, plus 2 f of x sub 2, and so on until we reach 2f of x sub n minus 1, plus f of x of n.

00:41 So for n equal to 4, the integral from 0 to 2 of f of x dx, this is approximately 1 half of delta x is b minus a over n times.

00:57 Well, we can change n to 4, and then you have f of x of 0 plus 2f of x of 1, plus 2 f of x of 2, plus 2 f of x of 3, plus f of x of 4.

01:16 Based on the table, this is 1 over 8 times b minus a is 2 minus 0 times f of 0 plus 2 ,000, times f of 0 plus 2, of 0 .5 plus 2f of 1 .0 plus 2 f of 1 .5 plus f of 2 of 2.

01:41 2 that will be 1 4th times f of 0 which is 3 plus f of 0 .5 that's 3 times 2 will be 6 plus 2 times f of 1 which is 5 so that's 2 times 5 or 10 plus 2 times h 16 plus f of 2 which is 13.

02:04 This gives us a value equal to 1 over 4 times 48 or 12.

02:12 And now for number 13, we let f of x equal to x cubed plus 3x squared minus 9x plus 7.

02:24 And this problem we want to determine the x values where this function is increasing.

02:29 So when we say increasing, it's when the first derivative is greater than zero.

02:35 So our first step will be to find the derivative of f.

02:39 Now, f prime of x by the power rule will be equal to 3x squared plus 6x minus 9.

02:47 And then from here we want to find the critical numbers of the function.

02:52 So we want to set this to 0.

02:54 We have 3x squared plus 6x minus 9 equals 0.

02:59 We factor at the 3, we get 3 times x squared plus 2x minus 3 equals 0.

03:05 We factor that even further.

03:08 We have 3 times x plus 3 times x minus 1 equals 0, which gives us x equals negative 3 and x equals 1.

03:18 Next, we want to create a number line from negative infinity.

03:25 To infinity and then you plug in or we plot the critical numbers in this number line, have your negative 3 and then 1.

03:33 So these will be our possible intervals...

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