answer:To solve for the value of a, we first need to understand the sets A and B and their intersection A cap B. 1. **Understanding Set A:** - The set A is defined as A={x|x^{2}-4leqslant 0}. This inequality can be rewritten as -2leqslant xleqslant 2 because it represents the range of x values for which x^2 is less than or equal to 4. Thus, A={x | -2leqslant xleqslant 2}. 2. **Understanding Set B:** - The set B is defined as B={x | 2x+aleqslant 0}. Solving this inequality for x gives xleqslant -frac{1}{2}a. Thus, B={x|xleqslant -frac{1}{2}a}. 3. **Finding the Intersection A cap B:** - The intersection A cap B is given as {x | -2leqslant xleqslant 1}. This means that the upper limit of set B must be 1 to match the intersection, leading to the equation -frac{1}{2}a=1. 4. **Solving for a:** - From the equation -frac{1}{2}a=1, we solve for a: [ -frac{1}{2}a = 1 implies a = -2. ] Therefore, the value of a that satisfies the given conditions is -2. Thus, the correct answer is boxed{text{B}}.

answer:The derivative of y=xe^x+x+1 is y'=(1+x)e^x+1, At the point (0,1), the slope of the tangent line is 1+1=2, Hence, the equation of the tangent line at the point (0,1) is y=2x+1. Thus, the answer is: boxed{2x-y+1=0}. To find the equation of the tangent line, we first find the derivative of the function y. The slope of the tangent line at the given point is equal to the derivative at that point. Using the point-slope form of a linear equation, we can find the equation of the tangent line.

answer:Since sin ^{2}theta + cos ^{2}theta = 1, we obtain the following system of equations: begin{cases} sin theta + cos theta = sqrt {2} sin ^{2}theta + cos ^{2}theta = 1 end{cases} Now, solve this system of equations for sin theta and cos theta. This will give us: sin theta = frac{sqrt{2}}{2}, cos theta = frac{sqrt{2}}{2} Thus, we have tan theta = frac{sin theta}{cos theta} = 1. Now, to find tan (theta + frac{pi}{3}), we use the tangent addition formula: tan (theta + frac{pi}{3}) = frac{tan theta + tan frac{pi}{3}}{1 - tan theta tan frac{pi}{3}} Plugging in the values, we get: tan (theta + frac{pi}{3}) = frac{1 + sqrt{3}}{1 - (1)(sqrt{3})} = boxed{-2 - sqrt{3}}

answer:1. **Understanding the rotation**: A rotation of 90^circ counter-clockwise about the origin transforms each point (x, y) to (-y, x). Using the rotation matrix for 90^circ, which is: [ begin{bmatrix} 0 & -1 1 & 0 end{bmatrix} ] 2. **Applying the rotation to y = log_2 x**: Each point (x, log_2 x) on the graph transforms to (-log_2 x, x) upon rotation. 3. **Expressing new coordinates**: The new x-coordinate is -log_2 x, and the new y-coordinate is x. To find an expression in terms of the new x-coordinate, we need: [ y = 2^{log_2 x} Rightarrow y = 2^{-text{new } x}. ] **Conclusion**: The equation for the new graph G' is y = 2^{-x}. y = 2^{-x} The final answer is boxed{A}