EmSAT Achieve Math Practice 1

what is the fourth derivative of of shown below at equals 0 of equals 2x plus 1 to the power 4. so let me show you the derivatives so this is the first derivative and this is the second third and fourth derivative so to find the fourth derivative as here in the question we're going to solve the first second third and fourth so let me write formula which is of equals two plus one to the power of 4. so as we can see here in the formula we have function inside larger function so that's mean we're going to use chain rule so what does chain rule says derivative of outside function living inside function alone so we have the derivative outside function and leaving the function inside alone then times the derivative of inside function so we have here power so if we have here power we're going to use the power rule so what does the power rule says for example here have 2x to the power of 3 so three divided by two as the power rule it will be six then three minus one equals two so we're going to do three times two then we're going to minus it by one so we'll do the same here so we here we have power four so i'm going to put it here then minus one so the first derivative equals so moved before here then as we said we're going to leave inside function as it is at the chain rule so i'm going to leave inside function as it is and then 4 minus 1 equals 3 then the derivative inside so let's derivative inside so 2x plus one so when we see without any power it's mean the number inside front of the is the derivative how's that so if don't have power mean to the power of 1. so 1 times equals 2 then 1 minus 1 is 0 and to the power 0 is 1. so we don't need to write it and then here 1 without any it's mean constant number so any constant number and the the derivative of it will be zero so here we have the derivative is two so now we need to simplify so to simplify we're going to do 2 times 4 equals then here have nothing to divide with it so i'm gonna leave it as it is so now find the first derivative let me find the second derivative so the second derivative equals we're going to do the same way so here the power so 3 times 8 equals and then as the chain rule we're going to leave the number here minus 1 the number inside as it is then the times the derivative inside so going to put the number inside as it is then 3 minus 1 equals 2 and then the derivative inside as we solved it before it's 2 and then again we're going to simplify so 2 times 24 equals 48 and then i'm going to leave the number here as it is find the first derivative and we find the second derivative and this was the answer for the second derivative as we solved it and now we're going to find the third then the fourth derivative so let me write the third derivative equals so we're going to use the chair rule again so let me remind you of the chain rule again so derivative of outside which is this one function leaving inside function as it is then times the derivative inside so the derivative outside this one so we have power so i'm going to use the power rule so 2 times 48 here minus 1 that's the power rule so for 2 times 48 equals 96 then the number here as it is as the chain rule so i'm going to leave the function inside as it is so 2x plus 1 then 2 minus 1 equals 1 then the derivative inside so as said here we have if without any power mean the number directly in front of and here we have without we have number without which means constant number so it's zero so here two and now i'm going to simplify so two times 96 equals 192 then here i'm going to leave it as it is 2x plus 1 to the power of 1 because have nothing to divide to multiply it with and now we find the third and now we're going to go to the fourth derivative so the fourth derivative equals we're going to do the same way so 1 times 192 equals 192 and then here minus 1 so i'm going to leave the function as it is as said up for the general one and then one minus one equals zero then the derivative inside which is which we found up is two and now let's solve it so i'm going to keep 190 as it is you could multiply it now or you could multiply it later but i'm going to multiply it later so here we have two plus one so let me tell you about something so here if we have for example five to the power of zero it's directly one anything with the power of zero for example one to the power of zero it's always one so here we have two plus one so if it's power to the zero it's going to be directly one so anything with power of 0 means the derivative will be 1 and then i'm going to leave 2 as it is and equals i'm going to put 100 192 as it is so 2 times 1 equals 2 then times 2 equals 384 so the fourth derivative equals 384 diagram belongs to circle with the radius and ob the measure of angle aop is 120 degrees and the length of the radius is 6 centimeter which expression represents the length of the arc and centimeters so we can see here we have curve so the curve here is known as the arc length the arc of and now what we need to do is to use the formula of the arc length so what does the formula says equals so what does mean is the arc length it's the distance between point point and point around the edge of the circle so time equals theta so theta is the central angle in radians times which is the distance between and as we can see here so now what we need to do is to convert theta from degrees to radians to do that we will multiply it by pi divided by 180 so we will divide it by pi over so it will be theta equals 120 degrees times by over 180 so it's we have here 120 times by over 180 so here we can cancel the zeros so have to cancel here zero and here zero so if we cancel the zeros it will be twelve over eighteen and here we have pi so twelfth by so twelfth here is also known as six times two and the eighteen is also known as six times three so six times two equals twelve and i'm gonna leave the by here and then over 6 times 3 which is 18 and now we can cancel the 6 as we can see here we have 6 up and 6 down so theta i'm gonna write it up theta equals 2 by as we can see here two by left up over what's left down is only three so two by over three now we will find the arc length so i'm gonna write it here so as said the arc length formula is so is the arc length equals theta times so here we find theta which is 2 pi over 3 so over three times which is the point between and as said so it's six and now we will multiply 6 by 2 which will give us and then i'm gonna keep the pi as it is so as here multiply it by two and here we have over so the three here it will be over one because multiplied here so it's going to be one over three 12 pi so which answer is it it's this one suppose that for certain city in the uae in any given year the probability of sun storm is 0.15 and the probability of rain is 0.08 and the probability of both sun storm and rain is 0.02 what is the probability of rain given that sun storm hits so we have here the st sandstorm is 0.15 so i'm gonna call it and then the rain the probability of the rain is 0.08 so i'm gonna call it and then the probability of both is and so to find the probability need to use the conditional probability formula so the formula says probability be given equals probability and over probability of so we just need to replace the numbers here in the formula so as we we solved here so we can see that 0.15 and 0.08 is and and is 0.02 so here we need the probability of and so the probability of and is 0.02 so 0.02 and the probability of is 0.15 so let's check which answer is it so it's going to be this one