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(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola.

104.

Question

For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?

105.

Question

Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 – 3x. Find the values of a, b, c, and d.

106.

Question

Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x – 2.

107.

Question

Where is the function h(x) = |x – 1| + |x + 2| differentiable? Give a formula for h’ and sketch the graphs of h and h’.

108.

Question

(a) For what values of x is the function f(x) = |x2 – 9| differentiable? Find a formula for f’.

(b) Sketch the graphs of f and f’.

109.

Question

Find a parabola with equation y = ax2 + bx + c that has slope 4 at x = 1, slope –8 at x = –1, and passes through the point (2, 15).

110.

Question

Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (–2, 6) and (2, 0).

111.

Question

The equation y” + y’ – 2y = x2 is called a differential equation because it involves an unknown function y and its derivativesy’ and y”. Find constants A, B and C such that thefunction y = Ax2 + Bx + C satisfies this equation. (Differentialequations will be studied in detail in Chapter 9.)

112.

Question

Find a second-degree polynomial P such that P(2) = 5, P'(2) = 3, and P” = 2.

113.

Question

Find nth the derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

(a) f (x) = xn (b) f (x) = 1/x

114.

Question

Use the definition of a derivative to show that if f (x) = 1/x, then f ‘(x) = –1/x2. (This proves the Power Rule for the case n = –1.)

115.

Question

(a) Find equations of both lines through the point (2, –3) that are tangent to the parabola y = x2 + x.

(b) Show that there is no line through the point (2, 7) that is tangent to the parabola. Then draw a diagram to see why.
116.

Question

Where does the normal line to the parabola y = x – x2 at the point (1, 0) intersect the parabola a second time? Illustrate with a sketch.

117.

Question

Find an equation of the normal line to the parabola y = x2 – 5x + 4 that is parallel to the line x – 3y = 5.
118.

Question

At what point on the curve y = 1 + 2ex – 3x is the tangent line parallel to the line 3x – y = 5? Illustrate by graphing the curve and both lines.

119.

Question

Find equations of both lines that are tangent to the curve y = 1 + x3 and parallel to the line 12x – y = 1.

120.

Question

Find an equation of the tangent line to the curve y = x vx that is parallel to the line y = 1 + 3x.