
Math
Hello! This is a good question, but the answer is not complicated. Division is by definition operation inverse to multiplication. a/b = c by definition means that a = b*c. Now about division by...

Math
There are 27 students in the class. The maximum number of groups of 4 is 6, since 6x4=24 and 7x4=28. (There cannot be 7 groups as there aren't enough students, but there are 6 groups.) Once 6...

Math
Determine if the sequence `a_n=(1)^n9^n` converges or diverges. The first few terms are 9,81,729,6561,... This sequence is divergent since it is neither bounded below or above. (There are terms...

Math
A right cone, is a solid of revolution generated by the rotation of a right triangle, around one leg. The circle described by the other leg is called the base of the cone and the upper end of the...

Math
There is a mathematical formula which will produce the number of combinations, or permutations, from a number of selections. Note that combinations and permutations have different mathematical...

Math
Hello! This all isn't very complicated, but requires some work. Probably you know that if we take the same number (x) several times (n) and have to sum them up, then the result is x*n (x multiplied...

Math
The average weight is given as 22lbs with a population standard deviuation of 3lbs. Assuming a normal distribution of weights, what is the weight at the 70th percentile? We use a standard normal...

Math
Let's denote the number of downloads of the standard version by x and the number of downloads of the highquality version by y. Since yesterday the highquality version was downloaded four times as...

Math
Hello! In general, the answer is "yes". Unfortunately your question isn't very specific, so I'll try to guess. If you mean linear functions of one variable, they have the form y=ax+b. If another...

Math
This function is differentiable everywhere. Therefore it reaches its absolute minimum and maximum either at the endpoints or where its derivative is zero. f(2) = 16  12 + 24 + 1 = 3, f(3) = 54...

Math
Given: `f(x)=x^36x^2+5,[3,5].` Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s). `f'(x)=3x^212x=0` `3x(x4)=0` `x=0,x=4` The critical...

Math
Given: `f(x)=3x^44x^312x^2+1,[2,3].` Find the critical numbers by setting the first derivative equal to zero and solving for the x value(s). `f'(x)=12x^312x^224x=0` `12x(x^2x2)=0`...

Math
f(x) is continuous and differentiable everywhere, so it reaches its maximum and minimum either at the endpoint or where f'(x)=0. f(1) = 0, f(2) = 27. `f'(x) = 3(x^21)^2*2x` . This is zero at x=0,...

Math
Given: `f(x)=x+(1/x),[0.2,4]` Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s). `f'(x)=1(1/x^2)=0` `1=(1/x^2)` `x^2=1`...

Math
Given: `f(x)=x/(x^2x+1), [0,3].` Find the critical values by setting the first derivative equal to zero and solving for the x values. Find the derivative using the quotient rule....

Math
Given: `f(t)=tsqrt(4t^2),[1,2].` Find the critical number by setting the first derivative equal to zero and solving for the t values. Find the derivative using the product rule....

Math
`f(t)=root(3)(8t)` differentiating, `f'(t)=(1/3)(8t)^(1/31)(1)` `f'(t)=1/(3(8t)^(2/3))` Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to...

Math
`f(t)=2cos(t)+sin(2t)` differentiating, `f'(t)=2sint+2cos(2t)` Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers...

Math
`f(t)=t+cot(t/2)` differentiating, `f'(t)=11/2csc^2(t/2)` Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers that are...

Math
`f(x)=xe^(x^2/8)` differentiating by applying product rule, `f'(x)=x(e^(x^2/8))((2x)/8)+e^(x^2/8)` `f'(x)=e^(x^2/8)(x^2/4+1)` `f'(x)=1/4e^(x^2/8)(x^24)` Now to find the absolute extrema of...

Math
This function is continuous on the given interval and is differentiable inside it. So it reaches minimum and maximum either at endpoint or where f'(x)=0. f(1/2) = 1/2ln(1/2)=1/2 + ln(2)....

Math
The function is defined because x^2+x+1 always >0. Further, ln(y) is monotone increasing and therefore it has min and max where y has min and max. x^2+x+1 has a global minimum at x=1/2 and it...

Math
`f(x)=x2tan^1(x)` differentiating, `f'(x)=12/(x^2+1)` `f'(x)=(x^2+12)/(x^2+1)` `f'(x)=(x^21)/(x^2+1)` Now to find the absolute extrema of the function , that is continuous on a closed...

Math
Given: `g(x)=x^(1/3)x^(2/3)` Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s). `g'(x)=(1/3)x^(2/3)+(2/3)x^(5/3)=0`...

Math
Given: `F(x)=x^(4/5)(x4)^2` Find the critical number(s) by setting the first derivative equal to zero and solve for the x value(s). `F'(x)=x^(4/5)[2(x4)]+(x4)^2[(4/5)x^(1/5)]=0`...

Math
You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation `g'(theta) = 0` . You need to evaluate the first derivative: `g'(theta) = 4 ...

Math
You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation `f'(theta) = 0` . You need to evaluate the first derivative: `f'(theta) = 2 sin...

Math
You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation h'(t) = 0. You need to evaluate the first derivative: `h'(t) = 3  1/(sqrt(1t^2))`...

Math
You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that: `f'(x) = 0` You need to find the first derivative using the...

Math
You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation f'(x) = 0. You need to evaluate the first derivative, using the product rule: `f'(x)=...

Math
Given: `f(x)=12+4xx^2,[0,5]` Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s). `f'(x)=42x=0` `4=2x` `2=x` The value x=2...

Math
Given: `f(x)=5+54x2x^3,[0,4]` Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s). `f'(x)=546x^2=0` `54=6x^2` `9=x^2` `x=3,x=3` The...

Math
`f(x) = 4 +(1/3)x  (1/2)x^2` `or, f(x) = 24 + 2x  3x^2` `or, f'(x) = 2  6x` Thus, for critical numbers f'(x) = 0 or, 26x = 0 or, x = 1/3 is the critical number

Math
You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that: `f'(x) = 0` `f'(x) = (x^3 + 6x^2  15x)` `f'(x) = 3x^2 + 12x...

Math
You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that: `f'(x) = 0` `f'(x) = (2x^3  3x^2  36x)` `f'(x) = 6x^2  6x...

Math
You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that: `f'(x) = 0` `f'(x) = (2x^3 + x^2 + 2x)` `f'(x) = 6x^2 + 2x +...

Math
`g(t) = t^4 + t^3 + t^2 + 1` `or, g'(t) = 4t^3 + 3t^2 + 2t` `or, g'(t) = t(4t^2 + 3t + 2)` ` g'(t) = 0` `(4t^2 + 3t + 2)t = 0` Hence the critical number is t = 0 ``

Math
The modulus of the equation of the function is defined as it follows: `3t  4 = 3t  4 if 3t  4>=0 => t >= 4/3` or `3t  4 = 4  3t if t < 4/3` You need to evaluate the critical...

Math
You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation g'(y) = 0. You need to evaluate the first derivative, using the quotient rule:...

Math
Given: `h(p)=(p1)/(p^2+4)` Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s). Find the derivative by using the quotient rule....

Math
Given: `h(t)=t^(3/4)2t^(1/4)` Find the critical number(s) by setting the first derivative equal to zero and solving for the x value(s). `h'(t)=(3/4)t^(1/4)(1/4)(2)t^(3/4)=0`...

Math
This function is defined for x>0. When x tends to +0, sqrt(x) tends to 0 and 1/sqrt(x) tends to +infinity, and y tends to +infinity. The same answer for x tending to infinity. And for x tending...

Math
`lim_(x>0)sqrt(9x^2)` plug in the value of x to evaluate the limit = `sqrt(90^2)=3` `lim_(x>oo)sqrt(9x^2)` =`sqrt(9oo^2)` limit does not exist

Math
You need to evaluate the limit, hence, you need to replace `oo` for x in equation: `lim_(x>oo) x(sqrt(1x^2))= (oo)(sqrt(1oo))` Since the result is indeterminate, you need to multiply and...

Math
`lim_(x>0)3xsin^2x` plug in the value of x `= 3*0sin^2(0)` =0 `lim_(x>oo) 3xsin^2x` `lim_(x>oo) x(3(sin^2x)/x)` Apply the squeeze theorem to evaluate the limit of sin^2x/x...

Math
`lim_(x>oo)sec^2(x/(x^2+1))` `lim_(x>oo)sec^2(x)/(x^2(1+1/x^2))` `lim_(x>oo)sec^2(1/(x(1+1/x^2)))` plug in the value of x to evaluate the limit `= sec^2(1/(oo(1+0)))` `= sec^2(0)=`...

Math
For a function y=f(x) the differential dy is f'(x)dx. Here `dy = 6x*dx` .

Math
You need to evaluate the limit, hence, you need to replace `oo` for x in equation: `lim_(x>oo) 3x^(2/3) = 3*root(3)(oo^2) = oo` Hence, evaluating the given limit, for `x>oo` , yields...

Math
If you evaluate the limit of function from the left, as `x > 0^` yields: `lim_(x>0^) csc(2x) = lim_(x>0^) 1/(sin(2x))` Replacing `2sinx*cosx` for `sin(2x)` yields: `lim_(x>0^)...

Math
`lim_(x>oo) (x+1)/(2x1)` `or, lim_(x>oo){(x/x)+(1/x)}/{((2x)/x)(1/x)}` `or, lim_(x>oo)(1+(1/x))/(2(1/x)) = 1/2` ``