Exam 1

  1. Question

    What is the derivative of f(x)=x8e3.4xf(x) = x^{8} e^{3.4 x}, evaluated at x=0.8x = 0.8?


    1. 25.4725.47
    2. 34.1334.13
    3. 38.7638.76
    4. 28.0228.02
    5. 28.9728.97

    Solution

    Using the product rule for f(x)=g(x)h(x)f(x) = g(x) \cdot h(x), where g(x):=x8g(x) := x^{8} and h(x):=e3.4xh(x) := e^{3.4 x}, we obtain f(x)=[g(x)h(x)]=g(x)h(x)+g(x)h(x)=8x81e3.4x+x8e3.4x3.4=e3.4x(8x7+3.4x8)=e3.4xx7(8+3.4x). \begin{aligned} f'(x) &= [g(x) \cdot h(x)]' = g'(x) \cdot h(x) + g(x) \cdot h'(x) \\ &= 8 x^{8 - 1} \cdot e^{3.4 x} + x^{8} \cdot e^{3.4 x} \cdot 3.4 \\ &= e^{3.4 x} \cdot(8 x^7 + 3.4 x^{8}) \\ &= e^{3.4 x} \cdot x^7 \cdot (8 + 3.4 x). \end{aligned} Evaluated at x=0.8x = 0.8, the answer is e3.40.80.87(8+3.40.8)=34.127595. e^{3.4 \cdot 0.8} \cdot 0.8^7 \cdot (8 + 3.4 \cdot 0.8) = 34.127595. Thus, rounded to two digits we have f(0.8)=34.13f'(0.8) = 34.13.


    1. False
    2. True
    3. False
    4. False
    5. False