a) Find dydx , when x = 2a , y = a . [2] b) Evaluate: (3 x 4)^2 x \, dx [2] c) Examine whether the function f(x) = 15x^2 14x + 1 is increasing or decreasing at x = 25 and x = 52 . [2]

a) Find $\frac{dy}{dx}$, when $x=2a\tan \theta$, $y=a\sec \theta$.

Step 1: Express $\frac{dy}{dx}$ in terms of $\frac{dy}{d\theta }$ and $\frac{dx}{d\theta }$

We are given: $$x=2a\tan \theta ,y=a\sec \theta$$ To find $\frac{dy}{dx}$, we use the chain rule: $$\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$$

Step 2: Compute $\frac{dx}{d\theta }$

Differentiate $x=2a\tan \theta$ with respect to $\theta$: $$\frac{dx}{d\theta }=2a\cdot \frac{d}{d\theta }(\tan \theta )=2a{\sec }^2\theta$$

Step 3: Compute $\frac{dy}{d\theta }$

Differentiate $y=a\sec \theta$ with respect to $\theta$: $$\frac{dy}{d\theta }=a\cdot \frac{d}{d\theta }(\sec \theta )=a\sec \theta \tan \theta$$

Step 4: Substitute into $\frac{dy}{dx}$

$$\frac{dy}{dx}=\frac{a\sec \theta \tan \theta }{2a{\sec }^2\theta }=\frac{\tan \theta }{2\sec \theta }$$

Step 5: Simplify using trigonometric identities

Recall that $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$. Thus: $$\frac{dy}{dx}=\frac{\frac{\sin \theta }{\cos \theta }}{2\cdot \frac{1}{\cos \theta }}=\frac{\sin \theta }{2}$$

Final Answer:

$$\boxed{\frac{dy}{dx}=\frac{\sin \theta }{2}}$$

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b) Evaluate: $\int (3\sin x-4{)}^2\cos xdx$

Step 1: Expand the integrand

First, expand $(3\sin x-4{)}^2$: $$(3\sin x-4{)}^2=9{\sin }^{2}x-24\sin x+16$$ Now multiply by $\cos x$: $$(9{\sin }^{2}x-24\sin x+16)\cos x=9{\sin }^{2}x\cos x-24\sin x\cos x+16\cos x$$

Step 2: Use substitution

Let $u=3\sin x-4$. Then: $$du=3\cos xdx⟹\cos xdx=\frac{du}{3}$$ Rewrite the integral in terms of $u$: $$\int (3\sin x-4{)}^2\cos xdx=\int u^2\cdot \frac{du}{3}=\frac{1}{3}\int u^{2}du$$

Step 3: Integrate

$$\frac{1}{3}\int u^{2}du=\frac{1}{3}\cdot \frac{u^3}{3}+C=\frac{u^3}{9}+C$$

Step 4: Substitute back $u=3\sin x-4$

$$\frac{(3\sin x-4{)}^3}{9}+C$$

Final Answer:

$$\boxed{\frac{(3\sin x-4{)}^3}{9}+C}$$

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c) Examine whether the function $f(x)=15x^2-14x+1$ is increasing or decreasing at $x=\frac{2}{5}$ and $x=\frac{5}{2}$.

Step 1: Find the derivative $f^{\prime }(x)$

Differentiate $f(x)$ with respect to $x$: $$f^{\prime }(x)=\frac{d}{dx}(15x^2-14x+1)=30x-14$$

Step 2: Evaluate $f^{\prime }(x)$ at $x=\frac{2}{5}$ and $x=\frac{5}{2}$

Final Answer:

• At $x=\frac{2}{5}$, the function is decreasing.
• At $x=\frac{5}{2}$, the function is increasing.