determine the number of solution for each of the following equations(where x is in radians): cosx=x/10

视频信息

答案文本 复制

视频字幕 复制

We need to find the number of solutions to the equation cosine x equals x over 10. To solve this graphically, we look for intersection points between the cosine function and the linear function y equals x over 10. Since cosine has range from negative 1 to 1, any solution must satisfy negative 10 less than or equal to x less than or equal to 10. To find the solutions systematically, we define h of x equals cosine x minus x over 10. The solutions are the zeros of this function. We evaluate h at key points to identify sign changes. At x equals negative 10, h is positive. At negative 3 pi, h becomes negative, indicating a sign change and thus a root between negative 10 and negative 3 pi. Continuing the sign analysis, we evaluate h at negative pi, pi, 2 pi, 3 pi, and 10. Each sign change between consecutive points indicates exactly one root in that interval. We identify seven intervals with sign changes, giving us seven roots total. The yellow dots show the approximate locations of these seven solutions. To verify our result, we analyze the derivative h prime of x equals negative sine x minus one tenth. This confirms that each interval with a sign change contains exactly one root. Therefore, the equation cosine x equals x over 10 has exactly seven solutions in the interval from negative 10 to 10. We need to determine how many solutions exist for the equation cosine x equals x over 10, where x is measured in radians. This is a transcendental equation that requires careful analysis. To solve this equation graphically, we plot both functions: y equals cosine x and y equals x over 10. The cosine function oscillates between negative 1 and positive 1, while x over 10 is a straight line passing through the origin with slope 1 over 10. The solutions are the x-coordinates where these graphs intersect. We use an analytical approach by defining h of x equals cosine x minus x over 10. The solutions to our original equation occur where h of x equals zero. Since cosine x is bounded between negative 1 and positive 1, we only need to consider x values between negative 10 and positive 10. We evaluate h of x at critical points to identify sign changes. At x equals negative 10, h is positive. At negative 3 pi and negative 2 pi, h remains positive. At negative pi, h becomes negative, indicating a sign change. At zero, h is positive again. At pi, h is negative. At 2 pi, h is positive, and at 3 pi, h is negative again. Each sign change indicates the presence of a root. In summary, we solved the equation cosine x equals x over 10 by analyzing the function h of x equals cosine x minus x over 10. Through systematic sign analysis at key points, we identified seven intervals where the function changes sign, indicating seven roots. Each interval contains exactly one solution, confirmed by derivative analysis. Therefore, the equation cos x equals x over 10 has exactly seven solutions.