Sivakumar MATH 171
Example Sheet 9
1. (From an old 151 common exam) Find all possible values of the real number λfor which
y=eλx satisfies the differential equation y+y0=y00.
2. Find an equation of the tangent to the curve y=exthat passes through the origin.
3. Find an equation of the tangent to the curve y=e−xthat is perpendicular to the line 2x−y= 8.
4. Find all numbers xsuch that ln(x2−2x−2) ≤0.
5. Let Cdenote the graph of the function y= ln(x). Suppose P(x0, y0), x0>1, is a point on C.
Suppose that the tangent to Cat Pmeets the x-axis at the point Q, and that the normal to C
at Pintersects the x-axis at R. Show that the area of the triangle PQR is x0+1
x0(ln(x0))2
2.
6. Consider the parametric curve Cgiven by the following equations:
x(t) = ln t, y(t) = tet, t > 0.
(i) Find an equation of the tangent line to the curve at the point (0, e).
(ii) Obtain a Cartesian equation for the curve C, and use it to verify your answer to Part (i).
7. Let f(x) = ex+ ln x,x > 0. If gis the inverse function of f, find g0(e).
8. If fis a one-to-one, twice differentiable function with inverse function g, show that
g00(x) = −f00 (g(x))
[f0(g(x))]3.
9. Suppose fis a differentiable function and f(u)>0 for every real number u. Find the derivative
of the function
f(f(log2x)) + log2(f(f(x))) .
10. Consider the function f(x) = ln(x+√x2+ 1).
(i) Show that the domain of fis (−∞,∞).
(ii) Show that fis an odd function. (Recall that a function fis said to be odd if f(−x) =
−f(x).)
(iii) What is the range of f?
(iv) Show that fis a one-to-one function on (−∞,∞).
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