Chapter 3: Q 2. TB (page 298)

Evaluation in Leibniz notation: Given that $r = r (t), s = s (t)$ and $u = u (t)$ are functions of $t$ , answer each of the following.
$(i)$ If $\frac{d s}{d t} = 3 s^{2} - 4,$ find ${\frac{d s}{d t}}_{s = 2}$
$(i i)$ If $r^{2} \frac{d r}{d t} - 2 r = 0,$ find ${\frac{d r}{d t}}_{r = 3}$
$(i i i)$ If $4 = 2 u \frac{d u}{d t}$ andlocalid="1649780328523" $u = 2 + 3 t,$ find ${\frac{d u}{d t}}_{t = 4}$

Short Answer

Expert verified

Part $(i) {\frac{d s}{d t}}_{s = 2} = 8$

Part $(i i) {\frac{d r}{d t}}_{r = 3} = \frac{2}{3}$

Part $(i i i) {\frac{d u}{d t}}_{t = 4} = \frac{1}{7}$

Step by step solution

Part (i) Step 1. Definition.

Leibniz notation is a method for representing the derivative that uses the symbols $d x$ and $d y$ to designate infinitely small increments of $x$ and $y$ .

Part (i) Step 2. Explanation of part i

Given that $\frac{d s}{d t} = 3 s^{2} - 4$

Put $s = 2$

$\Rightarrow {\frac{d s}{d t}}_{s = 2} = 3 {(2)}^{2} - 4 = 8$

Part (ii) Step 1. Explanation of part ii

Given that $r^{2} \frac{d r}{d t} - 2 r = 0$

$\Rightarrow \frac{d r}{d t} = \frac{2}{r}$

Put $r = 3$

$\Rightarrow {\frac{d r}{d t}}_{r = 3} = \frac{2}{3}$

Part (iii) Step 4. Explanation of part iii.

Given that $4 = 2 u \frac{d u}{d t}$ and $u = 2 + 3 t$

$\Rightarrow 4 = 2 (2 + 3 t) \frac{d u}{d t}$

$\Rightarrow \frac{d u}{d t} = \frac{2}{2 + 3 t}$

Put $t = 4$

$\Rightarrow {\frac{d u}{d t}}_{t = 4} = \frac{2}{14} = \frac{1}{7}$

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Short Answer

Step by step solution

Part (i) Step 1. Definition.

Part (i) Step 2. Explanation of part i

Part (ii) Step 1. Explanation of part ii

Part (iii) Step 4. Explanation of part iii.

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Evaluation in Leibniz notation: Given that r=rt,s=stand u=utare functions of t, answer each of the following.iIf dsdt=3s2-4,find dsdts=2iiIf r2drdt-2r=0,find drdtr=3iiiIf4=2ududtandlocalid="1649780328523" u=2+3t,finddudtt=4

Short Answer

Step by step solution

Part (i) Step 1. Definition.

Part (i) Step 2. Explanation of part i

Part (ii) Step 1. Explanation of part ii

Part (iii) Step 4. Explanation of part iii.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Geometry

Decision Maths

Pure Maths

Statistics

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