Created by StemPy LabA function is continuous on the whole real line and satisfies Differentiating both sides turns this into a differential equation for , but take care. The right side contains , so differentiating it needs , and you must first make sure that is differentiable, at least for . Also ask whether the equation says anything about the value of at , where the factor on the right side vanishes. Continuity of at may matter in the end. Which of the following statements are true?
One or more options are correct. Tick all the correct ones: a wrong option costs marks, and a part-right answer earns part of the marks.
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← Let F(x) = ∫₀^(x²) sin√ t dt for every real x. The upper limit x² is the same for x and for -x, so F is an even function. Near t = 0 the integrand behaves like √ t, and that decides how fast F(x) goes to 0. Recall that √(x²) = |x|, not x, which matters as soon as x is negative. Two of the statements are about what happens very close to x = 0, from one side or from both sides, and one is about the sign of F far from 0. The substitution t = w² gives F in closed form, but most of the statements can be decided without it. Which of the following statements are true?The figure shows the curve y = eˣ and its tangent at the point where x = s, for some s > 0. The curve bends upwards, so the tangent lies below it. The curve, the tangent and the y-axis enclose a region, shaded in the figure. Let A(s) be its area. For larger s the tangent crosses the y-axis below the origin, but the region is still bounded by the same three lines and curves. Find A(s) as a formula in s, and then check each statement. Which of the following statements are true? →
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