﻿ Left Hand, Right Hand Derivatives and Parametric Curves Worksheet | Problems & Solutions

# Left Hand, Right Hand Derivatives and Parametric Curves Worksheet

Left Hand, Right Hand Derivatives and Parametric Curves Worksheet
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1.
Find $\frac{dy}{dx}$ for the parametric curve $x$ = 6$t$, $y$ = 10$t$2.
 a. $\frac{10}{3}$$t$ b. $t$ c. $\frac{1}{t}$ d. $\frac{3}{10}$($\frac{1}{t}$) e. 1

#### Solution:

x = 6t , y = 10t2
[Equations of the parametric curves.]

dxdt = 6
[Differentiate x with respect to t.]

dydt = 20t
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= 20t6 = 10 / 3t

dydx = 10 / 3t

2.
Find $\frac{dy}{dx}$ for the parametric curve $x$ = 5$t$2 - 6, $y$ = $t$ + 11.
 a. $\frac{1}{10}$ b. $\frac{1}{10t}$ c. 10$t$ d. $\frac{1}{t}$ e. $t$

#### Solution:

x = 5t2 - 6, y = t + 11
[Equations of the parametric curves.]

dxdt = 10t
[Differentiate x with respect to t.]

dydt = 1
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= 110t

dydx = 110t

3.
Find the slope of the parametric curve $x$ = 7cos $t$, $y$ = 7sin $t$.
 a. 7cos $t$ b. cot $t$ c. - tan $t$ d. - cot $t$ e. tan $t$

#### Solution:

x = 7cos t , y = 7sin t
[Equations of the parametric curves.]

dxdt = - 7sin t
[ Differentiate x with respect to t .]

dydt = 7cos t
[Differentiate y with respect to t .]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= 7cos t- 7sint = - cot t
[Simplify.]

Slope of the parametric curve is - cot t

4.
Find $\frac{dy}{dx}$ for the parametric curve $x$ = 2$e$8$t$, $y$ = $t$$e$8$t$.
 a. $\frac{1}{8{e}^{8t}}$ b. ${e}^{8t}$ c. ${e}^{8t}\left(1+8t\right)$ d. 1 + 8$t$ e. $\frac{1+8t}{16}$

#### Solution:

x = 2e8t, y = te8t
[Equations of the parametric curves.]

dxdt = 16e8t
[Differentiate x with respect to t.]

dydt = e8t + 8te8t
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

dydx = e8t+8te8t16e8t

= 1+8t16
[Simplify.]

dydx = 1+8t16

5.
Find $\frac{dy}{dx}$ for the parametric curve $x$ = $\frac{2}{t}$, $y$ = $t$ - 48.
 a. $\frac{1}{{t}^{2}}$ b. 2 c. - $\frac{1}{{t}^{2}}$ d. $t$2 e. - $\frac{1}{2}$ $t$2

#### Solution:

x = 2t, y = t - 48
[Equations of the parametric curves.]

dxdt = - 2t2
[Differentiate x with respect to t.]

dydt = 1
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= 1- 2t2 = - 1 / 2 t2
[Simplify.]

dydx = - 1 / 2 t2

6.
Find the slope of the parametric curve $x$ = $t$3 - 8$t$, $y$ = 5$t$2.
 a. $\frac{2t}{3t-1}$ b. $\frac{1}{3\left({t}^{2}-1\right)}$ c. $\frac{t}{{t}^{2}-1}$ d. 2$t$ e. $\frac{10t}{3{t}^{2}-8}$

#### Solution:

x = t3 - 8t, y = 5t2
[Equations of parametric curves.]

dxdt = 3t2 - 8
[Differentiate x with respect to t.]

dydt = 10t
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= 10t3t2-8

The slope of the parametric curve is 10t3t2-8

7.
Find the slope of the parametric curve $x$ = $a$sec 8$t$, $y$ = $b$tan 8$t$.
 a. $\frac{a}{b}$cos 8$t$ b. $\frac{a}{b}$sin 8$t$ c. d. $\frac{b}{a}$sec 8$t$ e. $\frac{b}{a}$cosec 8$t$

#### Solution:

x = asec 8t , y = btan 8t
[Equations of the parametric curves.]

dxdt = 8asec 8t tan 8t
[ Differentiate x with respect to t .]

dydt = 8bsec2 8t
[Differentiate y with respect to t .]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= bsec2 8ta(sec 8t)(tan 8t) = bacosec 8t
[Simplify.]

Slope of the parametric curve is bacosec 8t

8.
Find $\frac{dy}{dx}$ for the parametric curve $x$ = 9 - $t$2, $y$ = $\sqrt{7-{t}^{2}}$.
 a. 2$\sqrt{7-{t}^{2}}$ b. $\frac{t}{2\sqrt{7-{t}^{2}}}$ c. $\frac{1}{\sqrt{7-{t}^{2}}}$ d. $\frac{1}{2\sqrt{7-{t}^{2}}}$ e.

#### Solution:

x = 9 - t2, y = 7-t2
[Equations of the parametric curves.]

dxdt = - 2t
[Differentiate x with respect to t.]

dydt = - t7-t2
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= - t7-t2- 2t

= 127-t2
[Simplify.]

dydx = 127-t2

9.
Find the slope of the parametric curve $x$ = cos3 2$t$, $y$ = sin3 2$t$.
 a. - tan 2$t$ b. 6sin2 2$t$cos 2$t$ c. - cot 2$t$ d. tan 2$t$sin 2$t$ e. - tan 2$t$sin 2$t$

#### Solution:

x = cos3 2t , y = sin3 2t
[Equations of the parametric curves.]

dxdt = - 6cos2 2t sin 2t
[ Differentiate x with respect to t .]

dydt = 6sin2 2t cos 2t
[Differentiate y with respect to t .]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= 6sin2 2t cos 2t- 6cos2 2t sin 2t = - tan 2t
[Simplify.]

Slope of the parametric curve is - tan 2t

10.
Find $\frac{dy}{dx}$ for the parametric curve $x$ = $e$7$t$cos 9$t$, $y$ = $e$7$t$sin 10$t$.
 a. b. c. - cot 9$t$ d. e. tan 10$t$

#### Solution:

x = e7tcos 9t, y = e7tsin 10t
[Equations of the parametric curves.]

dxdt = e7t(7cos 9t - 9sin 9t)
[Differentiate x with respect to t.]

dydt = e7t(7sin 10t + 10cos 10t)
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= e7t(7sin 10t+10cos 10t)e7t(7cos 9t-9sin 9t)

= 7sin 10t+10cos 10t7cos 9t-9sin 9t
[Simplify.]

dydx = 7cos 10t+10sin 10t7cos 9t-9sin 9t