Question Paper of Calculus - Rates of Change

Question 1:

$I f 5 x = 4 t a n d 6 y = 5 t a n d t > 0, t h e n$

Question 2:

$p^{2} (\frac{p + 1}{p}) \leq 1$

Question 3:

$A m e l t i n g i c e b e r g i s d e c r e a \sin g i n v o l u m e a t t h e r a t e o f \frac{3 π}{16} m^{3} / h o u r, b u t i t i s a l w a y s c o n i c a l i n s h a p e a n d i t s s e m i v e r t i c a l a n g l e r e m a i n s c o n s \tan t .$

$I n i t i a l l y t h e i c e b e r g h a s a b a s e r a d i u s 30 m e t r e s a n d h e i g h t 40 m e t r e s a s s h o w n o n t h e d i a g r a m (N o t e : V = \frac{1}{3} {πr}^{2} h) (i) F i n d t h e r a t e a t w h i c h t h e h e i g h t i s d e c r e a \sin g w h e n t h e h e i g h t o f t h e b l o c k i s 9 m e t r e s . (i i) F i n d t h e r a t e a t w h i c h t h e b a s e a r e a i s d e c r e a \sin g w h e n t h e r a d i u s i s 12 m e t r e s .$

Question 4:

$I f N = 65 w h e n t = 0, t h e n t h e s o l u t i o n t o \frac{d N}{d t} 0.3 (N - 20) i s : (A) N = 20 + 45 e^{0.3 t} (B) N = 20 + 45 e^{- 0.3 t} (C) N = 20 - 45 e^{0.3 t} (D) N = 20 - 45 e^{- 0.3 t}$

Question 5:

$c) F u l l y f a c t o r i s e 6 x^{3} + 17 x^{2} - 4 x - 3 .$

Question 6:

$E v a l u a t e \lim_{x \to 0} \frac{\sin 3 x}{3 x} F i n d \frac{d}{d x} [\ln \sqrt{\frac{1 + x}{1 - x}}] E v a l u a t e \int_{- 3}^{3} \frac{d x}{x^{2} + 9} U s e t h e s u b s t i t u t i o n \sqrt{x} = u t o e v a l u a t e \int_{1}^{4} \frac{d x}{x + \sqrt{x}}$

Question 7:

$S o l v e f o r x : \frac{x - 3}{x - 5} \leq 6, x \neq 5$

Question 8:

$E v a l u a t e 2 \int_{0}^{\frac{π}{4}} \cos^{2} 4 x d x f i n d t h e g e n e r a l s o l u t i o n t o : \cos 5 θ - \cos 2 θ = 0$

Question 9:

$A c e r t a i n p a r t i c l e m o v e s a l o n g t h e x a x i s a c c o r d i n g t o t = 2 x^{2} - 5 x + 3 w h e r e x i s m e a s u r e d i n m e t r e s a n d t i n s e c o n d s . I n i t i a l l y t h e p a r t i c l e i s 1.5 m t o t h e r i g h t o f O a n d m o v i n g a w a y f r o m O . i) P r o v e t h a t t h e v e l o c i t y, v m s^{- 1}, i s g i v e n b y v = \frac{1}{4 x - 5} . i i) F i n d a n e x p r e s s i o n f o r t h e a c c e l e r a t i o n i n t e r m s o f x . i i i) F i n d t h e v e l o c i t y o f t h e p a r t i c l e w h e n t = 6 s e c o n d s .$

Question 10:

$F i n d \int 3 \cos^{2} 3 x d x . T w o l i n e s m a k e a n a n g l e o f 45 ° w i t h o n e a n o t h e r . I f o n e l i n e h a s a g r a d i e n t o f 2, w h a t a r e t h e p o s s i b l e g r a d i e n t s o f t h e o t h e r l i n e ? U s e t h e s u b s t i t u t i o n u = 2 x + 6 t o f i n d \int x \sqrt{2 x + 6} d x .$