1)A function g is defined for all real numbers and has the following property:
g(a+b) - g(a) = 4ab+2b2 find g'(x)
A)4
B)-4
C)2x^2
D)4x
E)does not exist
not so sure where to start
2)If d/dx[f(x)] = g(x) and d/dx[g(x)] = f(3x, then d^2/dx^2 [f(x^2)] is
A)4x^2 f(3x^2) + 2g(x^2)
B)f(3x^2)
C)f(x^4)
D)2xf(3x^2) + 2g(x^2)
E)2xf(3x^2)
i was able to get (A) but not sure if its correct
3)lim as h-> 0 3(1/2 +h)^5 - 3(1/2)^5/h
A)0
B)1
C)15/16
D)limit does not exist
E)limit cannot be determined
thanks for the helps guys. ill really appreciate it
1)
Recall the definition of the derivative.
g'(x) = lim(h->0) (g(x+h)-g(x))/h
using the information in the definition of g, we have
= lim(x->0) (4xh + 2h^2)/h = 4x+2h = 4x
so, D
2)
d/dx f(x^2) = 2x d/dx f(x^2) = 2x g(x^2)
d/dx 2x g(x^2) = 2g(x^2) + 2x f(3x^2)
so, D
3)
this limit is just 3*d/dx(x^5) at x=1/2
= 3(5x^4) at x=1/2
= 3(5/16)
= 15/16
so, C
1) To find g'(x), we need to differentiate the given function g(a+b) - g(a) = 4ab + 2b^2 with respect to x.
Let's consider g(a+x) - g(a):
g(a+x) - g(a) = 4ax + 2x^2.
Now, we can see that this expression is of the form g(u) - g(v), where u = a+x and v = a. To find g'(x), we need to differentiate g(u) - g(v) with respect to u.
Differentiating g(u) - g(v) = 4au + 2u^2 with respect to u gives us:
g'(u) - g'(v) = 4a + 4u.
Since u = a+x, we can rewrite the above equation as:
g'(a+x) - g'(a) = 4a + 4(a+x).
Now, we can see that this expression is of the form g(w) - g(z), where w = a+x and z = a. To find g'(x), we need to differentiate g(w) - g(z) with respect to w.
Differentiating g(w) - g(z) = 4aw + 4(a+x) with respect to w gives us:
g'(w) - g'(z) = 4a.
Since w = a+x, we can rewrite the above equation as:
g'(a+x) - g'(a) = 4a.
Therefore, g'(x) = 4a.
Since the function g is defined for all real numbers, a can be any real number. Therefore, g'(x) = 4x.
So, the correct answer is D) 4x.
2) To find d^2/dx^2 [f(x^2)], we need to apply the chain rule twice.
Let's start by finding df/dx:
df/dx = g(x).
Now, let's find dg/dx:
dg/dx = f'(3x) * 3.
To find d^2f/dx^2, we differentiate df/dx = g(x) with respect to x:
d^2f/dx^2 = dg/dx = f'(3x) * 3.
Since we need to find d^2/dx^2 [f(x^2)], we substitute x^2 for x in the above equation:
d^2/dx^2 [f(x^2)] = f'(3x^2) * 3.
Therefore, the correct answer is E) 2xf(3x^2).
3) Let's simplify the given expression:
lim(h->0) [3(1/2 + h)^5 - 3(1/2)^5]/h.
Expanding (1/2 + h)^5 using the binomial theorem, we have:
lim(h->0) [3(1/2 + h)(1/2 + h)(1/2 + h)(1/2 + h)(1/2 + h) - 3(1/2)^5]/h.
Simplifying further, we get:
lim(h->0) [3(1/2 + 5/2h + 10/4h^2 + 10/8h^3 + 5/16h^4 + 1/32h^5) - 3(1/2)^5]/h.
Expanding and simplifying, we get:
lim(h->0) [3/2 + (15/2)h + (15/4)h^2 + (5/4)h^3 + (5/32)h^4 + (1/32)h^5 - 3/32]/h.
Canceling out h in the numerator and denominator, we have:
lim(h->0) [3/2 + (15/2)h + (15/4)h^2 + (5/4)h^3 + (5/32)h^4 + (1/32)h^5 - 3/32].
Now, if we plug in h = 0, we get:
[3/2 + (15/2)*0 + (15/4)*0^2 + (5/4)*0^3 + (5/32)*0^4 + (1/32)*0^5 - 3/32].
Simplifying further, we get:
3/2 - 3/32.
Calculating the solution, we have:
48/32 - 3/32 = 45/32 = 1 13/32.
Therefore, the correct answer is C) 15/16.
I hope my answers put a smile on your face!
1) To find g'(x), we need to take the derivative of g(x) with respect to x. Since g(a+b) - g(a) = 4ab + 2b^2, we can rewrite it as:
g(a + b) = g(a) + 4ab + 2b^2.
Let's replace a+b with x and a with x-b:
g(x) = g(x - b) + 4(x - b)b + 2b^2.
Now we can differentiate both sides of the equation with respect to x:
g'(x) = g'(x - b) + 4b.
So, g'(x) = 4b.
Since b can be any real number, g'(x) can take any real number value. Therefore, the answer is (E) does not exist.
2) We are given that d/dx[f(x)] = g(x) and d/dx[g(x)] = f(3x). We need to find d^2/dx^2[f(x^2)].
Let's start by finding d/dx[f(x^2)]:
Using the chain rule, we have:
d/dx[f(x^2)] = f'(x^2) * d/dx[x^2] = f'(x^2) * 2x.
Now let's find d^2/dx^2[f(x^2)]:
d^2/dx^2[f(x^2)] = d/dx[f'(x^2) * 2x] = d/dx[f'(x^2)] * 2x + f'(x^2) * d/dx[2x].
Using the chain rule again, we have:
d/dx[f'(x^2)] = f''(x^2) * d/dx[x^2] = f''(x^2) * 2x.
Substituting this back into the equation, we get:
d^2/dx^2[f(x^2)] = f''(x^2) * 2x * 2x + f'(x^2)*2 = 4x^2f''(x^2) + 2f'(x^2).
Now we are given that d/dx[g(x)] = f(3x). Substituting this into our equation, we get:
d^2/dx^2[f(x^2)] = 4x^2f''(x^2) + 2f'(x^2) = 4x^2[f(3x)] + 2g(x^2).
Therefore, the answer is (A) 4x^2f(3x) + 2g(x^2).
3) To find the limit as h approaches 0 of 3(1/2 + h)^5 - 3(1/2)^5/h, we can simplify the expression first.
The expression can be rewritten as:
3(1/2 + h)^5 - 3(1/2)^5/h = 3(1/2^5 + 5(1/2)^4h + 10(1/2)^3h^2 + 10(1/2)^2h^3 + 5(1/2)h^4 + h^5) - 3(1/2)^5/h.
Expanding and simplifying:
= 3(1/32 + 5/32h + 10/32h^2 + 10/32h^3 + 5/32h^4 + h^5) - 3(1/32)/h
= 3/32 + 15/32h + 30/32h^2 + 30/32h^3 + 15/32h^4 + 3h^5 - 3/32h.
Now, let's find the limit as h approaches 0:
lim(h->0) (3/32 + 15/32h + 30/32h^2 + 30/32h^3 + 15/32h^4 + 3h^5 - 3/32h)
= 3/32 + 0 + 0 + 0 + 0 + 0 - 0
= 3/32.
Therefore, the answer is (C) 15/16.
Hope this helps!
1) To find g'(x), we need to differentiate the given function g(a+b) - g(a) = 4ab+2b^2 with respect to x. Since g(a+b) and g(a) both depend on x, we need to apply the chain rule.
Let's differentiate both sides of the equation:
d/dx[(g(a+b) - g(a))] = d/dx[4ab+2b^2]
Using the chain rule, the left side becomes:
[d/d(a+b) * (g(a+b) - g(a))] * (da+db)/dx = [dg/da * (da+db)] * (da+db)/dx = dg/dx
The right side remains the same after differentiating with respect to x.
So, we have:
dg/dx = 4ab+2b^2
Since a and b are dummy variables, we can replace them with x:
dg/dx = 4xb + 2b^2
Now we need to simplify this expression. Since it's not explicitly given what function g(x) is, we can't simplify further without additional information. Therefore, we cannot find g'(x) without knowing more about the function.
Hence, the answer is E) does not exist.
2) To find d^2/dx^2[f(x^2)], we need to first differentiate f(x^2) with respect to x, and then differentiate the resulting expression with respect to x again.
We know that d/dx[f(x)] = g(x), so we can substitute g(x) into our expression:
d/dx[f(x^2)] = d/dx[f(x^2)] * d/dx[x^2] = g(x^2) * 2x
Now let's differentiate this expression with respect to x to get the second derivative:
d^2/dx^2[f(x^2)] = d/dx[g(x^2) * 2x]
= d/dx[g(x^2)] * 2x + g(x^2) * d/dx[2x]
Using the chain rule, we know that d/dx[g(x^2)] = g'(x^2) * d/dx[x^2]. Also, d/dx[2x] = 2.
Substituting these values back into the equation, we have:
d^2/dx^2[f(x^2)] = g'(x^2) * 2x + g(x^2) * 2
Now, to simplify further, let's substitute g(x^2) back with f(x^2) since we had the relation d/dx[f(x)] = g(x):
d^2/dx^2[f(x^2)] = g'(x^2) * 2x + f(x^2) * 2
Hence, the correct answer is D) 2xf(3x^2) + 2g(x^2).
3) To evaluate the limit lim h->0 [3(1/2 + h)^5 - 3(1/2)^5]/h, we can make use of the binomial expansion.
Expanding (1/2 + h)^5 using the binomial theorem, we get:
(1/2 + h)^5 = 1/32 + 5h/32 + 10h^2/32 + 10h^3/32 + 5h^4/32 + h^5/32
Now, let's substitute this expansion into the limit expression:
lim h->0 [3(1/2 + h)^5 - 3(1/2)^5]/h
= lim h->0 [3(1/32 + 5h/32 + 10h^2/32 + 10h^3/32 + 5h^4/32 + h^5/32) - 3(1/32)]/h
Simplifying this expression, notice that all the terms with powers of h greater than or equal to 2 contain at least one factor of h in the numerator.
Taking the limit as h approaches 0, those terms will all evaluate to zero. Only the term with h/32 remains:
lim h->0 [3(5h/32)]/h = lim h->0 [15h/32]/h = lim h->0 15/32 = 15/32
Therefore, the correct answer is C) 15/16.
Hope this helps! Let me know if you have any further questions.