A buffer consists of 0.20 M propanoic acid with Ka= 1.4 x 10^-5 and 0.30 M sodium propanoate. Calculate the pH of this buffer solution and determine if it is basic or acidic.
First, you need to recognize what contributes as the conjugate acid and conjugate base. The 0.20 M propanoic acid is the acid, as it says in the name. The 0.30 M sodium propanoate contributes the conjugate base because the sodium dissociates from the conjugate base ion in aqueous solutions. Second, you need to know that you can use the Henderson-Hasselbalch equation which is pH= pKa + log( [conj base] / [conj acid] ). We know the concentrations of the acid and the base. [conj base]= 0.30M and [conj acid]= 0.20M. The p in front of the Ka means take the -log of the Ka, or the acid dissociation constant. pKa= -log(1.4x10^-5) = 4.8539 pH= 4.8539 + log(0.3/.2) pH= 5.03 A pH below 7 is acidic so this buffer solution is slightly acidic.
For the function f(x)= x³+4x²+1, where the function is both increasing and concave up? What are the point(s) of inflection?
To start, you will need to take the first derivative of this function. This helps you determine where the function is decreasing or increasing. At values where f'(x) is positive, f(x) is increasing. At values where f'(x) is negative, f(x) is decreasing. Use the power rule to take the first derivative. f'(x)= 3x²+8x Factor to find the zeros of this function. f'(x)= x(3x+8) The zeros will be at 0 and (-8/3), and use these to make a sign chart. Test acceptable values for x in each case to find out if f'(x) is + or -. For x<(-8/3), f'(x) is positive. For (-8/3)<x<0, f'(x) is negative. For x>0, f'(x) is positive. From this information we can see that f(x) is increasing for x<(-8/3) and x>0. F(x) is decreasing between (-8/3) and 0. Use the power rule again to take second derivative. f"(x)=6x+8 Set equal to 0 to find the zero of the function to be -4/3. For x<(-4/3), f"(x) is negative and for x>(-4/3), f"(x) is positive. This means that f(x) is concave down for x<(-4/3) and concave up for x>(-4/3). Part of the question asks for point(s) of inflection. These are where the graph changes concavity. So the only point of concavity is x=(-4/3) Solving where f(x) is both increasing and concave up combines the two parts above. The answer has to be x>0 or x<(-8/3) from part one. The only interval that has numbers in the same bounds in part two as in part one is x>(-4/3). We cannot include the numbers from (-4/3) to 0 though because that would not be an area where the function is increasing. For this reason, the interval where f(x) is increasing AND concave up is x>0.
If f(x)=3x³ - 7x + 5, then f(-1)= ?
First, you should recognize that f(-1) is asking you to plug in -1 for x into the equation that you are given. Now, you can figure out what each separate term would equal and add or subtract the numbers accordingly. Plug in -1 to 3x³ 3(-1)³ the parenthesis are important because you need to remember that the negative sign goes with the 1. When solving, you need to remember PEMDAS. There is an exponent so you need to do that first. (-1)³= -1 So now you have 3(-1)= -3 Next is the 7x 7(-1)= -7 Lastly is +5 Substitute back in the values and do not forget about any original signs in the function. f(-1)= -3 - (-7) + 5. Here recognize that - (-7) is really adding 7 -3 + 7 +5 = 9 f(-1)=9