If f^{prime}(x)=a sin x+b cos x,f^{prime}(0)= | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given,$f^{\prime}(x)=a \sin x+b \cos x$ and $f^{\prime}(0)=4$.
Substituting $x=0$ in the derivative: $f^{\prime}(0)=a \sin(0)+b \cos(0) = b = 4$.

Vedclass · Accepted Answer

$2 \cos x+4 \sin x+1$

Vedclass · Answer

(B) Given,$f^{\prime}(x)=a \sin x+b \cos x$ and $f^{\prime}(0)=4$.
Substituting $x=0$ in the derivative: $f^{\prime}(0)=a \sin(0)+b \cos(0) = b = 4$.
Now,integrate $f^{\prime}(x)$ to find $f(x)$:
$f(x) = \int (a \sin x + 4 \cos x) dx = -a \cos x + 4 \sin x + C$.
Using $f(0)=3$: $f(0) = -a \cos(0) + 4 \sin(0) + C = -a + C = 3 \Rightarrow C = a+3$.
Using $f\left(\frac{\pi}{2}\right)=5$: $f\left(\frac{\pi}{2}\right) = -a \cos\left(\frac{\pi}{2}\right) + 4 \sin\left(\frac{\pi}{2}\right) + C = 0 + 4(1) + C = 4+C = 5 \Rightarrow C = 1$.
Substituting $C=1$ into $C=a+3$: $1 = a+3 \Rightarrow a = -2$.
Thus,$f(x) = -(-2) \cos x + 4 \sin x + 1 = 2 \cos x + 4 \sin x + 1$.

If $f^{\prime}(x)=a \sin x+b \cos x$,$f^{\prime}(0)=4$,$f(0)=3$ and $f\left(\frac{\pi}{2}\right)=5$,then $f(x)=$

Similar Questions

Find the following integral:
$\int \left(x^{\frac{3}{2}} + 2e^{x} - \frac{1}{x}\right) dx$

$\int \frac{\text{cosec}\theta - \cot\theta}{\text{cosec}\theta + \cot\theta} \, d\theta = $

$\int \sqrt{1+\cos x} \, dx$ is equal to

If $\int \frac{\sqrt{x}}{\sqrt{x}-\sqrt[3]{x}} d x=x+E x^{5 / 6}+D x^{2 / 3}+C x^{1 / 2}+B x^{1 / 3}+A x^{1 / 6}+\log (\sqrt[6]{x}-1)^6+K$,then $A+B+C+D+E=$

$\int \frac{2}{1+x+x^2} d x=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module