What is the equation of the tangent line to the curve f(x) = 3x^2 + 4x - 7 at the point where x = 2?
To come up with the equation of the tangent line, we would need a point on the line and the slope of the line. With it, we could write the equation using the point-slope form of a straight line. The point on the line can be found by substituting x = 2 into the original equation. This will give us the corresponding y coordinate. To find the slope of the line, we would take the derivative of the equation, then substitute x = 2 into the new equation. f(2) = 3(2)^2 + 4(2) -7 = 13 f'(x) = 6x + 4 f'(2) = 6(2) + 4 = 16 The point on the line would be (2, 13). The slope of the line is 16. Using the point slope form of y-y1 = m(x-x1) , the final solution would be y-13 = 16(x-2)
Assume the heights of dogs are approximately normally distributed. If the mean height of dogs are 3 feet tall and the standard deviation is 1 feet, about what proportion of dogs will be shorter than 4 feet tall?
We would first normalize the problem using the equation z = (x-m)/std where m stands for mean and std stands for standard deviation. The z value for a height of 4 feet would be z = (4-3)/1 = 1. This tells us that a height of 4 feet is 1 standard deviation above the mean. We can approximate the proportion using the 68-95-99.7 rule. 68% of the population falls within one standard deviation away from the mean, 95% within two, and 99.7% within three. Using the symmetry of a normal distribution, we can already determine that half of the population would be less than 3 feet tall. The remaining population of dogs between 3 and 4 feet would therefore have to be half of 68% or 34% because they only represent from the mean to one standard deviation above the mean. The total proportion less than 4 feet tall would be the sum of 50% and 34% or 84%.
70 is 35% of what number?
The best way to approach a problem such as this one would be to translate the word problem into a mathematical expression. "Is" is equals and "of" is times. The number we are looking for is the unknown and should be written as x. Percentages represent proportions or decimals so 35% should be divided by 100 to get the proper decimal value. This changes the problem to become: 70 = 0.35 * x To isolate for x, we would divide both sides by the coefficient of 0.35. Dividing 70 by 0.35 gives us the solution of 200.