# Tutor profile: Aishwarya S.

## Questions

### Subject: Calculus

Proof \lim_{\Theta \rightarrow 0} \frac{1 - cos \Theta }{\Theta } = 0

We have to set the limit so the when theta approaches 0 the limit doesn't exist, so with the help of algebraic manipulation to the equation. We will add {1 + cos \Theta} to the numerator and the denominator, this way we are not changing the equation and also restricting denominator theta to equal 0. we get \lim_{\Theta \rightarrow 0} \frac{1 - cos \Theta }{\Theta } * {\frac{1 + cos \Theta}{1 + cos \Theta}} = 0 when we simplify the equation we get = \lim_{\Theta \rightarrow 0} \frac{1 - cos^{2} \Theta }{\Theta (1 + cos \Theta)} According to the Pythagorean trig identity 1 - cos \Theta = sin^{2} \Theta therefore after substituting this identity, we get, = \lim_{\Theta \rightarrow 0} \frac{sin^{2} \Theta}{\Theta (1 + cos \Theta)} to simplify the equation more we will rewrite the trig identity sin^{2} \Theta = sin \Theta * sin \Theta now we get, = \lim_{\Theta \rightarrow 0} \frac{sin \Theta * sin \Theta }{\Theta (1 + cos \Theta)} = \lim_{\Theta \rightarrow 0} \frac{sin \Theta}{\Theta} * \frac{sin \Theta}{(1 + cos \Theta)} = \lim_{\Theta \rightarrow 0} \frac{sin \Theta}{\Theta} * \lim_{\Theta \rightarrow 0} \frac{sin \Theta}{(1 + cos \Theta)} From squeeze theorem, we know that lim_{\Theta \rightarrow 0} \frac{sin \Theta}{\Theta} = 1 therefore, = 1 * \lim_{\Theta \rightarrow 0} \frac{sin \Theta}{(1 + cos \Theta)} = \lim_{\Theta \rightarrow 0} \frac{sin \Theta}{(1 + cos \Theta)} If you see that when theta approaches 0 sin \Theta = 0 cos \Theta = 1 = \frac{0}{2} = 0 hence, \lim_{\Theta \rightarrow 0} \frac{1 - cos \Theta }{\Theta } = 0

### Subject: Physics

A baseball is popped straight upwards at 25.3 m/s. Determine the highest point of the flight of the ball and how long it will take to reach its highest point

To determine the highest point the baseball can reach can use the following kinematic formula: v_{f}^{2} = v_{i}^{2} + 2a\Delta x v_{i} = 25.3 \frac{m}{s} (the initial speed of the ball) v_{f} = 0 \frac{m}{s} (the final speed of the ball at the higest point) a = -9.8 \frac{m}{s^{2}} (the negative force the gravity will be applied on the ball, due to netown third law of motion) \Delta x = (the max distance the ball will travel) if you rearrange the kinetic formula to isolate for distance \Delta x = \frac{v_{f}^{2} - v_{i}^{2}}{2a} \Delta x = \frac{0^{2} - 25.3^{2}}{2(-9.8)} \Delta x = 32.66 m therefore, the highest point the ball will reach is 32.66m with the initial speed of 25.3 \frac{m}{s} To find how long it will take the ball to reach its highest point with the following kinematic equation: \Delta x = (\frac{v_{i} + v_{f}}{2}) * t rearrange the formula to isolate for time t = \frac{2\Delta x} {{v_{i} + v_{f}}} t = \frac{2*32.66} {{25.3 + 0}} t = 2.58s therefore, it took about 2.58s for the ball to reach its highest point with the initial speed of 25.3 \frac{m}{s}

### Subject: Algebra

Solve the following equation using the quadratic formula -x^2+8x =1

The above-mentioned equation can be solved using the quadratic formula. First, let's see the equation in the base form of the equation; ax^{2}+ bx + c = 0 where a, b and c are coefficients. let's write the above equation in this format; a= -1 (x^{2} term), b= 8(x term) and c = -1(constant term) ( you may think its 0 but if you see the base form the equation is equal to 0. so we have the take the (1) to the other side of the equation and equally the whole thing to 0. when you take (1) to the other side it becomes (-1)). Now that we have the base form set, let's look at the quadratic formula. Quadratic Formula is x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} so if we substitute the terms in the quadratic formula we will able to find the real solution of the equation. x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} x=\frac{-8\pm\sqrt{8^2-(4*(-1)*(-1))}}{2*(-1)} x=\frac{-8\pm\sqrt{64-(4)}}{-2} x=\frac{-8\pm\sqrt{60}}{-2} if you solve the above formula by adding and subtracting (sqrt60) to (-8) and dividing it to (-2) you will get 2 solution for this equation. x= 7.87 and x = 0.127

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