Interpreting the derivative Find the derivati...

Question

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

Suppose the speed of a car approaching a stop sign is given by v (t) = (t-5)², for 0 ≤ t ≤ 5, where t is measured in seconds and v(t) is measured in meters per second. Find v′(3).

Accepted Answer

Welcome back, everyone. The temperature of a cooling object is modeled by T of T equals T minus 6 to the power of T, where T is the time in minutes and capital T of T is the temperature in degrees Celsius. Calculate T at 5 and interpret its physical meaning, including units. So we're given our original function, capital T of T equals T minus 62, right? And this is temperature, right? So we can say temperature as a function of time. So temperature in this case is measured in degrees Celsius versus time. Which is measured in minutes. Now if we evaluate the derivative of capital T, we have to recall that it represents the rate of change. First of all, we're going to evaluate the derivative, and this means that we're evaluating D divided by the T of T minus 6 squad. And here we can apply the chain rule. First of all, we have an exponents. We apply the power rule. We bring down the 2. This gives us t minus 6 to the power of T minus 1. And then based on the chain rule, we have to multiply that by the derivative of the inner function T minus 6, which simply gives us 1, right, because the derivative of T is 1 based on the power rule and the derivative of -6 is 0. So overall, we get 2 multiplied by T minus 6, and this function shows us the rate of change of temperature in units of degrees Celsius per minute because that's how we measure time. And now what we want to do is evaluate T at 0.5, so that's 5 minutes after the point of origin, right? Essentially we are starting to measure temperature at time equals 0, and then in 5 minutes we want to evaluate the rate of change of temperature. So what we want to do is substitute 5 into the expression of the derivative, and we get 2 multiplied by 5 minus 6, which gives us 2 multiplied by -1, and that's -2, and the units would be degrees Celsius per minute. Now let's say that the interpretation of the statement would be the following. First of all, we're going to say that The temperature. And now we want to say, is it increasing or decreasing and because we have a negative sign for the rate of change, this means that it is decreasing, so the temperature is decreasing because of the negative sign. And now at what rate so it is decreasing at a rate. And now if we look at the numerical value, we're going to say of 2 °C per minute and now at what time, right at 5 because this is. The point of interest. So add 5 minutes because lower case t is measured in minutes and that's it. Our final answer is the temperature is decreasing at a rate of 2 °C per minute at 5 minutes. Thank you for watching.

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