# 4.8E: Exercises for Section 4.8

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In exercises 1 - 4, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

1) $$x=3+t,\quad y=1−t$$

2) $$x=8+2t, \quad y=1$$

$$m=0$$

3) $$x=4−3t, \quad y=−2+6t$$

4) $$x=−5t+7, \quad y=3t−1$$

$$m= -\frac{3}{5}$$

In exercises 5 - 9, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

5) $$x=3\sin t,\quad y=3\cos t, \quad \text{for }t=\frac{π}{4}$$

6) $$x=\cos t, \quad y=8\sin t, \quad \text{for }t=\frac{π}{2}$$

Slope$$=0; y=8.$$

7) $$x=2t, \quad y=t^3, \quad \text{for } t=−1$$

8) $$x=t+\dfrac{1}{t}, \quad y=t−\dfrac{1}{t}, \quad \text{for }t=1$$

Slope is undefined; $$x=2$$.

9) $$x=\sqrt{t}, \quad y=2t, \quad \text{for }t=4$$

In exercises 10 - 13, find all points on the curve that have the given slope.

10) $$x=4\cos t, \quad y=4\sin t,$$ slope = $$0.5$$

$$t=\arctan(−2); \left(\frac{4\sqrt{5}}{5},\frac{−8\sqrt{5}}{5}\right)$$.

11) $$x=2\cos t, \quad y=8\sin t,$$ slope= $$−1$$

12) $$x=t+\dfrac{1}{t}, \quad y=t−\dfrac{1}{t},$$ slope= $$1$$

No points possible; undefined expression.

13) $$x=2+\sqrt{t}, \quad y=2−4t,$$ slope= $$0$$

In exercises 14 - 16, write the equation of the tangent line in Cartesian coordinates for the given parameter $$t$$.

14) $$x=e^{\sqrt{t}}, \quad y=1−\ln t^2, \quad \text{for }t=1$$

$$y=−(\frac{2}{e})x+3$$

15) $$x=t\ln t, \quad y=\sin^2t, \quad \text{for }t=\frac{π}{4}$$

16) $$x=e^t, \quad y=(t−1)^2,$$ at $$(1,1)$$

$$y=2x−7$$

17) For $$x=\sin(2t), \quad y=2\sin t$$ where $$0≤t<2π.$$ Find all values of $$t$$ at which a horizontal tangent line exists.

18) For $$x=\sin(2t), \quad y=2\sin t$$ where $$0≤t<2π$$. Find all values of $$t$$ at which a vertical tangent line exists.

A vertical tangent line exists at $$t = \frac{π}{4},\frac{5π}{4},\frac{3π}{4},\frac{7π}{4}$$

19) Find all points on the curve $$x=4\cos(t), \quad y=4\sin(t)$$ that have the slope of $$\frac{1}{2}$$.

20) Find $$\dfrac{dy}{dx}$$ for $$x=\sin(t), \quad y=\cos(t)$$.

$$\dfrac{dy}{dx}=−\tan(t)$$

21) Find the equation of the tangent line to $$x=\sin(t), \quad y=\cos(t)$$ at $$t=\frac{π}{4}$$.

22) For the curve $$x=4t, \quad y=3t−2,$$ find the slope and concavity of the curve at $$t=3$$.

$$\dfrac{dy}{dx}=\dfrac{3}{4}$$ and $$\dfrac{d^2y}{dx^2}=0$$, so the curve is neither concave up nor concave down at $$t=3$$. Therefore the graph is linear and has a constant slope but no concavity.

23) For the parametric curve whose equation is $$x=4\cos θ, \quad y=4\sin θ$$, find the slope and concavity of the curve at $$θ=\frac{π}{4}$$.

24) Find the slope and concavity for the curve whose equation is $$x=2+\sec θ, \quad y=1+2\tan θ$$ at $$θ=\frac{π}{6}$$.

$$\dfrac{dy}{dx}=4, \quad \dfrac{d^2y}{dx^2}=−6\sqrt{3};$$ the curve is concave down at $$θ=\frac{π}{6}$$.

25) Find all points on the curve $$x=t+4, \quad y=t^3−3t$$ at which there are vertical and horizontal tangents.

26) Find all points on the curve $$x=\sec θ, \quad y=\tan θ$$ at which horizontal and vertical tangents exist.

No horizontal tangents. Vertical tangents at $$(1,0)$$ and $$(−1,0)$$.

In exercises 27 - 29, find $$d^2y/dx^2$$.

27) $$x=t^4−1, \quad y=t−t^2$$

28) $$x=\sin(πt), \quad y=\cos(πt)$$

$$d^2y/dx^2 = −\sec^3(πt)$$

29) $$x=e^{−t}, \quad y=te^{2t}$$

In exercises 30 - 31, find points on the curve at which tangent line is horizontal or vertical.

30) $$x=t(t^2−3), \quad y=3(t^2−3)$$

Horizontal $$(0,−9)$$;
Vertical $$(±2,−6).$$

31) $$x=\dfrac{3t}{1+t^3}, \quad y=\dfrac{3t^2}{1+t^3}$$

In exercises 32 - 34, find $$dy/dx$$ at the value of the parameter.

32) $$x=\cos t,y=\sin t, \quad \text{for }t=\frac{3π}{4}$$

$$dy/dx = 1$$

33) $$x=\sqrt{t}, \quad y=2t+4,t=9$$

34) $$x=4\cos(2πs), \quad y=3\sin(2πs), \quad \text{for }s=−\frac{1}{4}$$

$$dy/dx = 0$$

In exercises 35 - 36, find $$d^2y/dx^2$$ at the given point without eliminating the parameter.

35) $$x=\frac{1}{2}t^2, \quad y=\frac{1}{3}t^3, \quad \text{for }t=2$$

36) $$x=\sqrt{t}, \quad y=2t+4, \quad \text{for }t=1$$

$$d^2y/dx^2 = 4$$

37) Find intervals for $$t$$ on which the curve $$x=3t^2, \quad y=t^3−t$$ is concave up as well as concave down.

38) Determine the concavity of the curve $$x=2t+\ln t, \quad y=2t−\ln t$$.

Concave up on $$t>0$$.