Skip to main content

Subsection 2.11.2 Exercises

Exercises — Stage 1

1

If we implicitly differentiate \(x^2+y^2=1\text{,}\) we get the equation \(2x+2yy'=0\text{.}\) In the step where we differentiate \(y^2\) to obtain \(2yy'\text{,}\) which rule(s) below are we using? (a) power rule, (b) chain rule, (c) quotient rule , (d) derivatives of exponential functions

2

Using the picture below, estimate \(\ds\diff{y}{x}\) at the three points where the curve crosses the \(y\)-axis.

Remark: for this curve, one value of \(x\) may correspond to multiple values of \(y\text{.}\) So, we cannot express this curve as \(y=f(x)\) for any function \(x\text{.}\) This is one typical situation where we might use implicit differentiation.

3

Consider the unit circle, formed by all points \((x,y)\) that satisfy \(x^2+y^2=1\text{.}\)

  1. Is there a function \(f(x)\) so that \(y=f(x)\) completely describes the unit circle? That is, so that the points \((x,y)\) that make the equation \(y=f(x)\) true are exactly the same points that make the equation \(x^2+y^2=1\) true?
  2. Is there a function \(f'(x)\) so that \(y=f'(x)\) completely describes the slope of the unit circle? That is, so that for every point \((x,y)\) on the unit circle, the slope of the tangent line to the circle at that point is given by \(f'(x)\text{?}\)
  3. Use implicit differentiation to find an expression for \(\ds\diff{y}{x}\text{.}\) Simplify until the expression is a function in terms of \(x\) only (not \(y\)), or explain why this is impossible.

Exercises — Stage 2

4 (✳)

Find \(\ds\diff{y}{x}\) if \(xy + e^x + e^y = 1\text{.}\)

5 (✳)

If \(e^y=xy^2+x\text{,}\) compute \(\ds\diff{y}{x}\text{.}\)

6 (✳)

If \(x^2\tan(\pi y/4)+2x\log(y) = 16\text{,}\) then find \(y'\) at the points where \(y=1\text{.}\)

7 (✳)

If \(x^3+y^4 = \cos(x^2+y)\) compute \(\diff{y}{x}\text{.}\)

8 (✳)

If \(x^2e^y + 4x\cos(y) = 5\text{,}\) then find \(y'\) at the points where \(y=0\text{.}\)

9 (✳)

If \(x^2+y^2 = \sin(x+y)\) compute \(\diff{y}{x}\text{.}\)

10 (✳)

If \(x^2\cos(y)+2xe^y = 8\text{,}\) then find \(y'\) at the points where \(y=0\text{.}\)

11

At what points on the ellipse \(x^2+3y^2=1\) is the tangent line parallel to the line \(y=x\text{?}\)

12 (✳)

For the curve defined by the equation \(\sqrt{xy} = x^2y-2\text{,}\) find the slope of the tangent line at the point \((1, 4)\text{.}\)

13 (✳)

If \(x^2y^2+x\sin(y)=4\text{,}\) find \(\ds\diff{y}{x}\text{.}\)

Exercises — Stage 3

14 (✳)

If \(x^2+(y+1)e^y = 5\text{,}\) then find \(y'\) at the points where \(y=0\text{.}\)

15

For what values of \(x\) do the circle \(x^2+y^2=1\) and the ellipse \(x^2+3y^2=1\) have parallel tangent lines?