Find the equation of the tangent line to f(x) = x^2 + 3x + 15 at x=3
This is a VERY typical question asked on the AP Calculus AB and BC exams. Step 1: The first step to solving for any tangent line is to find the slope. Since we have the equation, we can solve for the derivative. The derivative will give us the instantaneous slope at x=3. f'(x)= 2x+3 Step 2: Plug in x=3 into the derivative to find the instantaneous slope at the coordinate x=3. f'(x)= 2x+3 f'(3)= 2(3)+3= 9 f'(x)=9 Thus, the slope is equal to 9. Step 3: Use the point-slope form to plug in what we have solved for and are given so far. (*m= slope) y-y1=m(x-x1) y-y1=9(x-3) In order to solve for y1, all we need to do is plug in x=3 into f(x). f(x) = x^2 + 3x + 15 f(3)=3^2+3(3)+15 = 33 y1=33 y-33=9(x-3) FINAL ANSWER (in point-slope): y-33=9(x-3) FINAL ANSWER (in slope-intercept form): y=9x+15
Researchers report that pools are unsafe to swim in if their chlorine level exceeds 3.0 parts per million (ppm) . What are the null and alternate hypotheses for this situation?
Step 1: Identify the population variable for means. μ= the true mean chlorine concentration in the water Step 2: Identify the null hypothesis. Remember, the null hypothesis ALWAYS involves an equal sign! Since this situation involves a testing for means, we MUST include units of measurement in our answer. Null Hypothesis: μ= 3 ppm Step 3: Identify the alternate hypothesis. Since we are testing to see whether or not the researchers are correct, we must identify the alternate hypothesis with a "greater than" sign (>). Alternate Hypothesis: μ> 3 ppm FINAL ANSWER: μ= the true mean chlorine concentration in the water Null Hypothesis: μ= 3 ppm Alternate Hypothesis: μ> 3 ppm
Solve for x: 6(x+2) - (x-4) = -5(x-4) +15
Step 1: When attempting to solve for "x", you must first multiply out the factors. 6x+ 12 - x + 4 = -5x + 20 + 16 Step 2: The next step is to combine like terms! 5x + 16 = -5x +36 Step 3: Gather like terms on one side of the equation in order to solve for x. Remember, what you do on one side of the equation must be applied to the other side! 5x + 16 = -5x +36 +5x +5x _______________ 10x +16= 36 -16 -16 _______________ 10x=20 Step 4: Now, divide each side by 10 in order to solve for x. FINAL ANSWER : x=2